NEW! Section 5.5 – The Substitution Rule
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.
- You can find additional practice problems for this section on the Practice Problems page for this Module.
Concepts
- The Substitution Rule
- The Substitution Rule for definite integrals
- Integrating symmetric functions
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 \(\displaystyle \int x(x^2+4)^5\,dx\)
\( \displaystyle \frac{1}{12} \left(x^2+4\right)^6 + C\)
2. Find \(\displaystyle{\int_0^{13}{\displaystyle{{2}\over{\sqrt[3]{(1+2x)^2}}}\,dx}}\)
\( \displaystyle 6\)
3. Find \(\displaystyle{\int{\displaystyle{{x^2}\over{(1-x)^4}}\,dx}}\)
\( \displaystyle \frac{1}{3(1-x)^3}-\frac{1}{(1-x)^2} +\frac{1}{1-x}+C\)
4. Find \(\displaystyle{ \int_{e^3}^{e^4} \displaystyle{{1}\over{x\ln x}}\,dx}\)
\( \displaystyle \ln \left(\frac{4}{3}\right)\)
5. Find \(\displaystyle \int \frac{4}{\arcsin(x)\sqrt{1-x^2}}\,dx\)
\( \displaystyle 4\ln \left| \arcsin (x)\right| + C\)
6. Find \(\displaystyle{\int_{0}^{\pi/4}e^{\sin (2t)}\cos (2t)\,dt}\)
\( \displaystyle \frac{1}{2}(e-1)\)
7. Find \(\displaystyle{\int\displaystyle{{x+1}\over{x^2+1}}\,dx}\)
\( \displaystyle \frac{1}{2}\ln \left(x^2+1\right) +\arctan (x)+C\)
8. Find \(\displaystyle{\int \displaystyle{{e^{5/x}}\over{x^2}}\,dx}\)
\( \displaystyle -\frac{1}{5}e^{\frac{5}{x}}+C\)