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
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 \(\displaystyle{\int 2xe^{5x}\,dx}\)
\( \displaystyle \frac{2}{5}xe^{5x}-\frac{2}{25}e^{5x}+C\)
2. Find \(\displaystyle{\int_0^1 x^2e^{-3x}\,dx}\)
\( \displaystyle -\frac{1}{3}e^{-3}-\frac{2}{9}e^{-3}-\frac{2}{27}e^{-3}+\frac{2}{27}\)
3. Find \(\displaystyle{\int (x-1)\cos(x)\,dx}\)
\( \displaystyle (x-1)\sin x + \cos x + C\)
4. Find \(\displaystyle{\int \displaystyle{{\ln x}\over{x^2}}\,dx}\)
\( \displaystyle -\frac{\ln x}{x} - \frac{1}{x}+C\)
5. Find \(\displaystyle{\int_1^9 \ln \sqrt{x}\,dx}\)
\( \displaystyle \frac{9}{2}\ln 9 -4\)
6. Find \(\displaystyle{\int \arcsin (2x)\,dx}\)
\( \displaystyle x\arcsin(2x)+\frac{1}{2}\sqrt{1-4x^2} +C\)
7. Find \(\displaystyle{\int_0^1 \arctan x\,dx}\)
\( \displaystyle \frac{\pi}{4}-\frac{1}{2}\ln 2\)
8. Find \(\displaystyle{\int x^5e^{x^3}\,dx}\)
\( \displaystyle \frac{1}{3}x^3e^{x^3}-\frac{1}{3}e^{x^3}+ C\)
9. Find \(\displaystyle{\int e^{5x} \sin (x)\,dx}\)
\( -\displaystyle \frac{1}{26} e^{5x}\cos x + \frac{5}{26}e^{5x}\sin x + C\)