{"id":3184,"date":"2024-06-13T17:26:14","date_gmt":"2024-06-13T17:26:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3184"},"modified":"2024-08-05T02:32:57","modified_gmt":"2024-08-05T02:32:57","slug":"antiderivatives-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/antiderivatives-fresh-take\/","title":{"raw":"Antiderivatives: Fresh Take","rendered":"Antiderivatives: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand indefinite integrals and learn how to find basic antiderivatives for functions<\/li>\r\n\t<li>Use the rule for integrating functions raised to a power<\/li>\r\n\t<li>Use antidifferentiation to solve simple initial-value problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Antiderivative<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li>Antidifferentiation is the reverse process of differentiation<\/li>\r\n\t<li>A function [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex] if [latex]F'(x) = f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">General Form of Antiderivatives:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex], then [latex]F(x) + C[\/latex] is the general form of all antiderivatives of [latex]f(x)[\/latex], where [latex]C[\/latex] is any constant<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Find all antiderivatives of [latex]f(x)= \\sin x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8800299\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8800299\"]<\/p>\r\n<p>What function has a derivative of [latex] \\sin x[\/latex]?<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"314667\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"314667\"]<\/p>\r\n<p>[latex]\u2212\\cos x+C[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the general antiderivative of [latex]f(x) = x\\sin(x) + x^2\\cos(x)[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"639907\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"639907\"]<\/p>\r\n<p>Let's approach this term by term:<\/p>\r\n<p>For [latex]x\\sin(x)[\/latex]:<\/p>\r\n<p>We can use integration by parts (which is the reverse of the product rule).<\/p>\r\n<p>Let [latex]u = x[\/latex] and [latex]dv = \\sin(x)dx[\/latex]<\/p>\r\n<p>Then [latex]du = dx[\/latex] and [latex]v = -\\cos(x)[\/latex]<\/p>\r\n<p>The antiderivative is [latex]uv - \\int v du = -x\\cos(x) + \\int \\cos(x)dx = -x\\cos(x) + \\sin(x)[\/latex]<\/p>\r\n<p>For [latex]x^2\\cos(x)[\/latex]:<\/p>\r\n<p>Again, use integration by parts<\/p>\r\n<p>Let [latex]u = x^2[\/latex] and [latex]dv = \\cos(x)dx[\/latex]<\/p>\r\n<p>Then [latex]du = 2xdx[\/latex] and [latex]v = \\sin(x)[\/latex]<\/p>\r\n<p>The antiderivative is [latex]x^2\\sin(x) - \\int 2x\\sin(x)dx[\/latex]<\/p>\r\n<p>For the remaining integral, use integration by parts again:<\/p>\r\n<p>[latex]\\int 2x\\sin(x)dx = -2x\\cos(x) + 2\\sin(x)[\/latex]<\/p>\r\n<p>Combining the results:<\/p>\r\n<p>[latex]F(x) = -x\\cos(x) + \\sin(x) + x^2\\sin(x) - (-2x\\cos(x) + 2\\sin(x)) + C[\/latex]<\/p>\r\n<p>Simplifying:<\/p>\r\n<p>[latex]F(x) = x^2\\sin(x) + x\\cos(x) - \\sin(x) + C[\/latex]<\/p>\r\n<p>Therefore, the general antiderivative is [latex]F(x) = x^2\\sin(x) + x\\cos(x) - \\sin(x) + C[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Indefinite Integrals<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Indefinite Integral:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]\\int f(x) dx = F(x) + C[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]C[\/latex] is the constant of integration<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Terminology:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int[\/latex] is the integral sign<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is the integrand<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex] is the variable of integration<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Properties:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sum\/Difference Rule: [latex]\\int (f(x) \\pm g(x)) dx = \\int f(x) dx \\pm \\int g(x) dx[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Constant Multiple Rule: [latex]\\int kf(x) dx = k\\int f(x) dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Power Rule for Integrals:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int x^n dx = \\frac{x^{n+1}}{n+1} + C[\/latex], for [latex]n \\neq -1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Common Indefinite Integrals:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{x} dx = \\ln|x| + C[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\sin x dx = -\\cos x + C[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\cos x dx = \\sin x + C[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\sec^2 x dx = \\tan x + C[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Verify that [latex]\\displaystyle\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/latex].<\/p>\r\n<p>[reveal-answer q=\"1770433\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"1770433\"]<\/p>\r\n<p>Calculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043257533\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043257533\"]<\/p>\r\n<p>[latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)= \\sin x+x \\cos x- \\sin x=x \\cos x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Evaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]<\/p>\r\n<p>[reveal-answer q=\"4078823\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4078823\"]<\/p>\r\n<p>Integrate each term in the integrand separately, making use of the power rule.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043259694\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043259694\"]<\/p>\r\n<p>[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the indefinite integral: [latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"39833\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"39833\"]<\/p>\r\n<p>Break down the integral using the sum\/difference rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx = \\int 3x^4 dx - \\int 2\\sin x dx + \\int 5e^x dx + \\int \\frac{4}{x} dx[\/latex]<\/p>\r\n<p>Apply the constant multiple rule:<\/p>\r\n<p style=\"text-align: center;\">[latex]= 3\\int x^4 dx - 2\\int \\sin x dx + 5\\int e^x dx + 4\\int \\frac{1}{x} dx[\/latex]<\/p>\r\n<p>Evaluate each integral:<\/p>\r\n<p style=\"text-align: center;\">[latex]<br \/>\r\n\\begin{array}{rcl}<br \/>\r\n3\\int x^4 dx &amp;=&amp; 3 \\cdot \\frac{x^5}{5} \\\\<br \/>\r\n-2\\int \\sin x dx &amp;=&amp; -2(-\\cos x) \\\\<br \/>\r\n5\\int e^x dx &amp;=&amp; 5e^x \\\\<br \/>\r\n4\\int \\frac{1}{x} dx &amp;=&amp; 4\\ln|x|<br \/>\r\n\\end{array}<br \/>\r\n[\/latex]<\/p>\r\n<p>Combine results and add the constant of integration:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{3x^5}{5} + 2\\cos x + 5e^x + 4\\ln|x| + C[\/latex]<\/p>\r\n<p>Therefore, [latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx = \\frac{3x^5}{5} + 2\\cos x + 5e^x + 4\\ln|x| + C[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Initial-Value Problems<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Definition of Differential Equation:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">An equation relating an unknown function and one or more of its derivatives<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Initial-Value Problem:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">A differential equation with an additional condition<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Condition typically specifies the function value at a particular point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Example:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = f(x), y(x_0) = y_0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solving Initial-Value Problems:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find the general solution of the differential equation<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the initial condition to determine the specific solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Motion problems (position, velocity, acceleration)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Growth and decay models<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Numerous real-world scenarios in physics, engineering, and other fields<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Suppose the car is traveling at the rate of [latex]44[\/latex] ft\/sec. How long does it take for the car to stop? How far will the car travel?<\/p>\r\n<p>[reveal-answer q=\"7660112\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"7660112\"]<\/p>\r\n<p>[latex]v(t)=-15t+44[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"923849\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"923849\"]<\/p>\r\n<p>[latex]2.93[\/latex] sec, [latex]64.5[\/latex] ft<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand indefinite integrals and learn how to find basic antiderivatives for functions<\/li>\n<li>Use the rule for integrating functions raised to a power<\/li>\n<li>Use antidifferentiation to solve simple initial-value problems<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Antiderivative<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li>Antidifferentiation is the reverse process of differentiation<\/li>\n<li>A function [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex] if [latex]F'(x) = f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">General Form of Antiderivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex], then [latex]F(x) + C[\/latex] is the general form of all antiderivatives of [latex]f(x)[\/latex], where [latex]C[\/latex] is any constant<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find all antiderivatives of [latex]f(x)= \\sin x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8800299\">Hint<\/button><\/p>\n<div id=\"q8800299\" class=\"hidden-answer\" style=\"display: none\">\n<p>What function has a derivative of [latex]\\sin x[\/latex]?<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q314667\">Show Solution<\/button><\/p>\n<div id=\"q314667\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\u2212\\cos x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the general antiderivative of [latex]f(x) = x\\sin(x) + x^2\\cos(x)[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q639907\">Show Answer<\/button><\/p>\n<div id=\"q639907\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s approach this term by term:<\/p>\n<p>For [latex]x\\sin(x)[\/latex]:<\/p>\n<p>We can use integration by parts (which is the reverse of the product rule).<\/p>\n<p>Let [latex]u = x[\/latex] and [latex]dv = \\sin(x)dx[\/latex]<\/p>\n<p>Then [latex]du = dx[\/latex] and [latex]v = -\\cos(x)[\/latex]<\/p>\n<p>The antiderivative is [latex]uv - \\int v du = -x\\cos(x) + \\int \\cos(x)dx = -x\\cos(x) + \\sin(x)[\/latex]<\/p>\n<p>For [latex]x^2\\cos(x)[\/latex]:<\/p>\n<p>Again, use integration by parts<\/p>\n<p>Let [latex]u = x^2[\/latex] and [latex]dv = \\cos(x)dx[\/latex]<\/p>\n<p>Then [latex]du = 2xdx[\/latex] and [latex]v = \\sin(x)[\/latex]<\/p>\n<p>The antiderivative is [latex]x^2\\sin(x) - \\int 2x\\sin(x)dx[\/latex]<\/p>\n<p>For the remaining integral, use integration by parts again:<\/p>\n<p>[latex]\\int 2x\\sin(x)dx = -2x\\cos(x) + 2\\sin(x)[\/latex]<\/p>\n<p>Combining the results:<\/p>\n<p>[latex]F(x) = -x\\cos(x) + \\sin(x) + x^2\\sin(x) - (-2x\\cos(x) + 2\\sin(x)) + C[\/latex]<\/p>\n<p>Simplifying:<\/p>\n<p>[latex]F(x) = x^2\\sin(x) + x\\cos(x) - \\sin(x) + C[\/latex]<\/p>\n<p>Therefore, the general antiderivative is [latex]F(x) = x^2\\sin(x) + x\\cos(x) - \\sin(x) + C[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<h2>Indefinite Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Indefinite Integral:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\int f(x) dx = F(x) + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]F(x)[\/latex] is an antiderivative of [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]C[\/latex] is the constant of integration<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Terminology:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int[\/latex] is the integral sign<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is the integrand<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex] is the variable of integration<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sum\/Difference Rule: [latex]\\int (f(x) \\pm g(x)) dx = \\int f(x) dx \\pm \\int g(x) dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Constant Multiple Rule: [latex]\\int kf(x) dx = k\\int f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Power Rule for Integrals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int x^n dx = \\frac{x^{n+1}}{n+1} + C[\/latex], for [latex]n \\neq -1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Indefinite Integrals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{x} dx = \\ln|x| + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\sin x dx = -\\cos x + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\cos x dx = \\sin x + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\sec^2 x dx = \\tan x + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Verify that [latex]\\displaystyle\\int x \\cos x dx=x \\sin x+ \\cos x+C[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q1770433\">Hint<\/button><\/p>\n<div id=\"q1770433\" class=\"hidden-answer\" style=\"display: none\">\n<p>Calculate [latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043257533\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043257533\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{d}{dx}(x \\sin x+ \\cos x+C)= \\sin x+x \\cos x- \\sin x=x \\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]\\displaystyle\\int (4x^3-5x^2+x-7) dx[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4078823\">Hint<\/button><\/p>\n<div id=\"q4078823\" class=\"hidden-answer\" style=\"display: none\">\n<p>Integrate each term in the integrand separately, making use of the power rule.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043259694\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043259694\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x^4-\\frac{5}{3}x^3+\\frac{1}{2}x^2-7x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the indefinite integral: [latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q39833\">Show Answer<\/button><\/p>\n<div id=\"q39833\" class=\"hidden-answer\" style=\"display: none\">\n<p>Break down the integral using the sum\/difference rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx = \\int 3x^4 dx - \\int 2\\sin x dx + \\int 5e^x dx + \\int \\frac{4}{x} dx[\/latex]<\/p>\n<p>Apply the constant multiple rule:<\/p>\n<p style=\"text-align: center;\">[latex]= 3\\int x^4 dx - 2\\int \\sin x dx + 5\\int e^x dx + 4\\int \\frac{1}{x} dx[\/latex]<\/p>\n<p>Evaluate each integral:<\/p>\n<p style=\"text-align: center;\">[latex]<br \/>  \\begin{array}{rcl}<br \/>  3\\int x^4 dx &=& 3 \\cdot \\frac{x^5}{5} \\\\<br \/>  -2\\int \\sin x dx &=& -2(-\\cos x) \\\\<br \/>  5\\int e^x dx &=& 5e^x \\\\<br \/>  4\\int \\frac{1}{x} dx &=& 4\\ln|x|<br \/>  \\end{array}<br \/>[\/latex]<\/p>\n<p>Combine results and add the constant of integration:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{3x^5}{5} + 2\\cos x + 5e^x + 4\\ln|x| + C[\/latex]<\/p>\n<p>Therefore, [latex]\\int (3x^4 - 2\\sin x + 5e^x + \\frac{4}{x}) dx = \\frac{3x^5}{5} + 2\\cos x + 5e^x + 4\\ln|x| + C[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<h2>Initial-Value Problems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Differential Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">An equation relating an unknown function and one or more of its derivatives<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Initial-Value Problem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A differential equation with an additional condition<\/li>\n<li class=\"whitespace-normal break-words\">Condition typically specifies the function value at a particular point<\/li>\n<li class=\"whitespace-normal break-words\">Example:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{dy}{dx} = f(x), y(x_0) = y_0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Initial-Value Problems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find the general solution of the differential equation<\/li>\n<li class=\"whitespace-normal break-words\">Use the initial condition to determine the specific solution<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Motion problems (position, velocity, acceleration)<\/li>\n<li class=\"whitespace-normal break-words\">Growth and decay models<\/li>\n<li class=\"whitespace-normal break-words\">Numerous real-world scenarios in physics, engineering, and other fields<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Suppose the car is traveling at the rate of [latex]44[\/latex] ft\/sec. How long does it take for the car to stop? How far will the car travel?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q7660112\">Hint<\/button><\/p>\n<div id=\"q7660112\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]v(t)=-15t+44[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q923849\">Show Solution<\/button><\/p>\n<div id=\"q923849\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2.93[\/latex] sec, [latex]64.5[\/latex] ft<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":32,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3184"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3184\/revisions"}],"predecessor-version":[{"id":3897,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3184\/revisions\/3897"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3184\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3184"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3184"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3184"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}