Problem 20

Question

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x \sin x d x $$

Step-by-Step Solution

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Answer

The integral of $x \sin x \, dx$ is $-x \cos x + \sin x$

1Step 1: Choose u and dv

Identify $u$ and $dv$ from the integral. In this case, we let $u = x$ and $dv = \sin x dx$.

2Step 2: Compute du and v

Calculate $du$ and $v$, the derivative of $u$ and the integral of $dv$, respectively. In this situation, $du = dx$ and $v = -\cos x$.

3Step 3: Apply Integration by Parts

Substitute $u$, $v$, $du$, and $dv$ into the integration by parts formula, $\int u dv = uv - \int v du$, to get the integral. This results in $-x \cos x + \int \cos x dx$.

4Step 4: Simplify the Integral

The remaining integral, $\int \cos x dx$, is simpler and can be readily integrated, resulting in $\sin x$. Thus, the solution to the original integral, $\int x \sin x dx$, becomes $-x \cos x + \sin x$.

Problem 20

Problem 21

Other exercises in this chapter

Problem 20

Find the integral. $$ \int \frac{t}{\left(1-t^{2}\right)^{3 / 2}} d t $$

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Problem 20

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x $$

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Problem 21

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{4}^{\infty} \frac{1}{x(\ln x)^{3}} d x $$

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Problem 21

Find the integral involving secant and tangent. $$ \int \sec ^{3} \pi x d x $$

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