What function has a derivative of [latex]\sin x[/latex]?

[latex]−\cos x+C[/latex]

Let’s approach this term by term:

For [latex]x\sin(x)[/latex]:

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

Let [latex]u = x[/latex] and [latex]dv = \sin(x)dx[/latex]

Then [latex]du = dx[/latex] and [latex]v = -\cos(x)[/latex]

The antiderivative is [latex]uv - \int v du = -x\cos(x) + \int \cos(x)dx = -x\cos(x) + \sin(x)[/latex]

For [latex]x^2\cos(x)[/latex]:

Again, use integration by parts

Let [latex]u = x^2[/latex] and [latex]dv = \cos(x)dx[/latex]

Then [latex]du = 2xdx[/latex] and [latex]v = \sin(x)[/latex]

The antiderivative is [latex]x^2\sin(x) - \int 2x\sin(x)dx[/latex]

For the remaining integral, use integration by parts again:

[latex]\int 2x\sin(x)dx = -2x\cos(x) + 2\sin(x)[/latex]

Combining the results:

[latex]F(x) = -x\cos(x) + \sin(x) + x^2\sin(x) - (-2x\cos(x) + 2\sin(x)) + C[/latex]

Simplifying:

[latex]F(x) = x^2\sin(x) + x\cos(x) - \sin(x) + C[/latex]

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

Calculate [latex]\frac{d}{dx}(x \sin x+ \cos x+C)[/latex].

[latex]\frac{d}{dx}(x \sin x+ \cos x+C)= \sin x+x \cos x- \sin x=x \cos x[/latex]

Integrate each term in the integrand separately, making use of the power rule.

[latex]x^4-\frac{5}{3}x^3+\frac{1}{2}x^2-7x+C[/latex]

Break down the integral using the sum/difference rule:

[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]

Apply the constant multiple rule:

[latex]= 3\int x^4 dx - 2\int \sin x dx + 5\int e^x dx + 4\int \frac{1}{x} dx[/latex]

Evaluate each integral:

[latex]
\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}
[/latex]

Combine results and add the constant of integration:

[latex]\frac{3x^5}{5} + 2\cos x + 5e^x + 4\ln|x| + C[/latex]

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]

[latex]v(t)=-15t+44[/latex]

[latex]2.93[/latex] sec, [latex]64.5[/latex] ft