Antiderivatives: Fresh Take

Lumen Learning

Antiderivatives: Fresh Take

Understand indefinite integrals and learn how to find basic antiderivatives for functions
Use the rule for integrating functions raised to a power
Use antidifferentiation to solve simple initial-value problems

Finding the Antiderivative

The Main Idea

Antidifferentiation is the reverse process of differentiation
A function $F(x)$ is an antiderivative of $f(x)$ if $F'(x) = f(x)$ for all $x$ in the domain of $f$
General Form of Antiderivatives:
- If $F(x)$ is an antiderivative of $f(x)$ , then $F(x) + C$ is the general form of all antiderivatives of $f(x)$ , where $C$ is any constant

Find all antiderivatives of $f(x)= \sin x$ .

What function has a derivative of $\sin x$ ?

$−\cos x+C$

Find the general antiderivative of $f(x) = x\sin(x) + x^2\cos(x)$ .

Let’s approach this term by term:

For $x\sin(x)$ :

We can use integration by parts (which is the reverse of the product rule).

Let $u = x$ and $dv = \sin(x)dx$

Then $du = dx$ and $v = -\cos(x)$

The antiderivative is $uv - \int v du = -x\cos(x) + \int \cos(x)dx = -x\cos(x) + \sin(x)$

For $x^2\cos(x)$ :

Again, use integration by parts

Let $u = x^2$ and $dv = \cos(x)dx$

Then $du = 2xdx$ and $v = \sin(x)$

The antiderivative is $x^2\sin(x) - \int 2x\sin(x)dx$

For the remaining integral, use integration by parts again:

$\int 2x\sin(x)dx = -2x\cos(x) + 2\sin(x)$

Combining the results:

$F(x) = -x\cos(x) + \sin(x) + x^2\sin(x) - (-2x\cos(x) + 2\sin(x)) + C$

Simplifying:

$F(x) = x^2\sin(x) + x\cos(x) - \sin(x) + C$

Therefore, the general antiderivative is $F(x) = x^2\sin(x) + x\cos(x) - \sin(x) + C$ .

Indefinite Integrals

The Main Idea

Definition of Indefinite Integral:
- Notation: $\int f(x) dx = F(x) + C$
- $F(x)$ is an antiderivative of $f(x)$
- $C$ is the constant of integration
Terminology:
- $\int$ is the integral sign
- $f(x)$ is the integrand
- $x$ is the variable of integration
Key Properties:
- Sum/Difference Rule: $\int (f(x) \pm g(x)) dx = \int f(x) dx \pm \int g(x) dx$
- Constant Multiple Rule: $\int kf(x) dx = k\int f(x) dx$
Power Rule for Integrals:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , for $n \neq -1$
Common Indefinite Integrals:
- $\int \frac{1}{x} dx = \ln|x| + C$
- $\int e^x dx = e^x + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int \sec^2 x dx = \tan x + C$

Verify that $\displaystyle\int x \cos x dx=x \sin x+ \cos x+C$ .

Calculate $\frac{d}{dx}(x \sin x+ \cos x+C)$ .

$\frac{d}{dx}(x \sin x+ \cos x+C)= \sin x+x \cos x- \sin x=x \cos x$

Evaluate $\displaystyle\int (4x^3-5x^2+x-7) dx$

$x^4-\frac{5}{3}x^3+\frac{1}{2}x^2-7x+C$

Evaluate the indefinite integral: $\int (3x^4 - 2\sin x + 5e^x + \frac{4}{x}) dx$

Break down the integral using the sum/difference rule:

$\int (3x^4 - 2\sin x + 5e^x + \frac{4}{x}) dx = \int 3x^4 dx - \int 2\sin x dx + \int 5e^x dx + \int \frac{4}{x} dx$

Apply the constant multiple rule:

$= 3\int x^4 dx - 2\int \sin x dx + 5\int e^x dx + 4\int \frac{1}{x} dx$

Evaluate each integral:

$\begin{array}{rcl} 3\int x^4 dx &=& 3 \cdot \frac{x^5}{5} \\ -2\int \sin x dx &=& -2(-\cos x) \\ 5\int e^x dx &=& 5e^x \\ 4\int \frac{1}{x} dx &=& 4\ln|x| \end{array}$

Combine results and add the constant of integration:

$\frac{3x^5}{5} + 2\cos x + 5e^x + 4\ln|x| + C$

Therefore, $\int (3x^4 - 2\sin x + 5e^x + \frac{4}{x}) dx = \frac{3x^5}{5} + 2\cos x + 5e^x + 4\ln|x| + C$

Initial-Value Problems

The Main Idea

Definition of Differential Equation:
- An equation relating an unknown function and one or more of its derivatives
Initial-Value Problem:
- A differential equation with an additional condition
- Condition typically specifies the function value at a particular point
- Example:
  - $\frac{dy}{dx} = f(x), y(x_0) = y_0$
Solving Initial-Value Problems:
- Find the general solution of the differential equation
- Use the initial condition to determine the specific solution
Applications:
- Motion problems (position, velocity, acceleration)
- Growth and decay models
- Numerous real-world scenarios in physics, engineering, and other fields

Suppose the car is traveling at the rate of $44$ ft/sec. How long does it take for the car to stop? How far will the car travel?

$v(t)=-15t+44$

$2.93$ sec, $64.5$ ft