Math 152: Worksheet 6

Integration by Parts

Learning Problems

These problems should be completed on your own. If you need hints on solving a problem, there are some provided with each problem. Click on the word "hint" to view it and again to hide it. They go in increasing order of helpfulness, with the last hint mostly giving away how to do the problem. Try to work from the earlier hints to the later ones, as this will give you the practice you need to succeed in this class.

Problem 1

Compute $\displaystyle \int x e^{5x}\ dx $

• Hint 1

  This is a product, so integration by parts seems to be the right way to go. What should you integrate and differentiate?

• Hint 2

  Try $u = x$ and $dv = e^{5x}\ dx$. What is the integral of the $dv$ term?

• Hint 3

  The integral of $e^{5x}$ is $\displaystyle \frac{1}{5}e^{5x}$

• Answer

  $\displaystyle \frac{1}{5}xe^{5x} - \frac{1}{25}e^{5x} + C$

Problem 2

Compute $\displaystyle \int x^{-5}\ln{x}\ dx$

• Hint 1

  This is a product, so integration by parts seems to be the right way to go. What should you integrate and differentiate?

• Hint 2

  Natural log is hard to integrate, so try $u = \ln{x}$ and $dv = x^{-5}\ dx$

• Hint 3

  After the first round of integration by parts, you should be left with just a power of $x$. This can just be integrated directly.

• Answer

  $\displaystyle -\frac{1}{4} \frac{\ln{x}}{x^4} - \frac{1}{16x^4} + C$

Problem 3

Compute $\displaystyle \int 3x \sin{(4-x)}\ dx$

• Hint 1

  This is a product, so integration by parts seems to be the right way to go. What should you integrate and differentiate?

• Hint 2

  Try $u = 3x$ and $dv = \sin(4-x)\ dx$. How do you integrate this $dv$ term?

• Hint 3

  The integral of $\sin(4-x)$ is $\cos(4-x) + C$.

• Answer

  $\displaystyle 3x \cos(4-x) + 3\sin(4-x) + C$

Problem 4

Compute $\displaystyle \int 4x \cos(x^2 + 5)\ dx$

• Hint 1

  This one is not integration by parts. What other technique do we have so far?

• Hint 2

  Try $u = x^2 + 5$ in a $u$ substitution

• Answer

  $\displaystyle 2\sin(x^2 + 5) + C$

Problem 5

Compute $\int_0^1 x \cos{\pi x}\ dx$

• Hint 1

  This is a definite integral, but the idea of the problem is still the same. This is a product, so what should you differentiate and integrate?

• Hint 2

  Try $u = x$ and $dv = \cos(\pi x)$.

• Answer

  $\displaystyle -\frac{2}{\pi^2}$

Problem 6

Compute $\displaystyle \int x^2 e^{2x}\ dx$

• Hint 1

  If at first you don't succeed...

• Hint 2

  This will require two rounds of integration by parts. Why do we know that? What's going to happen each time you do integration by parts?

• Hint 3

  You want to differentiate the polynomial part, so start with $u=x^2$. After the first round, you will want $u = x$.

• Answer

  $\displaystyle \frac{1}{2}x^2e^{2x} - \frac{1}{2}xe^{2x} + \frac{1}{4}e^{2x} + C $

Problem 7

Compute $\displaystyle \int \sin^{-1}(x)\ dx$

• Hint 1

  We do not have a formula for this. However, we can differentiate $\sin^{-1}(x)$. How can we use this to help us?

• Hint 2

  Try $u = \sin^{-1}(x)$ and $dv = dx$.

• Hint 3

  After the round of integration by parts, you should end up with $$ x \sin^{-1}(x) - \int \frac{x dx}{\sqrt{1-x^2}} $$

• Answer

  $\displaystyle x\sin^{-1}(x) + \sqrt{1-x^2} + C$

Submission Problems

Problem 1

Compute $\displaystyle \int \sqrt{x}e^{\sqrt{x}}\ dx$ using any method so far. Hint: Substitution then integration by parts.

Problem 2

Compute $\displaystyle \int_{\pi/4}^{\pi/3} \cos(x) \ln(\sin(x))\ dx$