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Virtual Math Learning Center

Virtual Math Learning Center Texas A&M University Virtual Math Learning Center

Section 7.2 - Integration by Parts

Instructions

  • The first video below explains the concepts in this section. 
  • This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.

Concepts

  • Using integration by parts to solve integrals
  • Using integration by parts multiple times and possibly having the original integral reappear in the solution

If you would like to see more videos on the topic, click the following link and check the related videos.


Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. Find \(\displaystyle{\int 2xe^{5x}\,dx}\)

\( \displaystyle \frac{2}{5}xe^{5x}-\frac{2}{25}e^{5x}+C\)

If you would like to see more videos on this topic, click the following link and check the related videos.

2. Find \(\displaystyle{\int_0^1 x^2e^{-3x}\,dx}\)

\( \displaystyle -\frac{1}{3}e^{-3}-\frac{2}{9}e^{-3}-\frac{2}{27}e^{-3}+\frac{2}{27}\)

If you would like to see more videos on this topic, click the following link and check the related videos.

3. Find \(\displaystyle{\int (x-1)\cos(x)\,dx}\)

\( \displaystyle (x-1)\sin x + \cos x + C\)

If you would like to see more videos on this topic, click the following link and check the related videos.

4. Find \(\displaystyle{\int \displaystyle{{\ln x}\over{x^2}}\,dx}\)

\( \displaystyle -\frac{\ln x}{x} - \frac{1}{x}+C\)

If you would like to see more videos on this topic, click the following link and check the related videos.

5. Find \(\displaystyle{\int_1^9 \ln \sqrt{x}\,dx}\)

\( \displaystyle \frac{9}{2}\ln 9 -4\)

If you would like to see more videos on this topic, click the following link and check the related videos.

6. Find \(\displaystyle{\int \arcsin (2x)\,dx}\)

\( \displaystyle x\arcsin(2x)+\frac{1}{2}\sqrt{1-4x^2} +C\)

If you would like to see more videos on this topic, click the following link and check the related videos.

7. Find \(\displaystyle{\int_0^1 \arctan x\,dx}\)

\( \displaystyle \frac{\pi}{4}-\frac{1}{2}\ln 2\)

If you would like to see more videos on this topic, click the following link and check the related videos.

8. Find \(\displaystyle{\int x^5e^{x^3}\,dx}\)

\( \displaystyle \frac{1}{3}x^3e^{x^3}-\frac{1}{3}e^{x^3}+ C\)

If you would like to see more videos on this topic, click the following link and check the related videos.

9. Find \(\displaystyle{\int e^{5x} \sin (x)\,dx}\)

\( -\displaystyle \frac{1}{26} e^{5x}\cos x + \frac{5}{26}e^{5x}\sin x + C\)

If you would like to see more videos on this topic, click the following link and check the related videos.