Math 152: Worksheet 1

The Substitution Method

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 $\displaystyle \int \cos(8x - 5)\ dx$

• Hint 1

  What is the inside function here? Try making that $u$.

• Hint 2

  Set $u = 8x - 5$. What is $du$? What does the integral become?

• Answer

  $\frac{1}{8}\sin(8x - 5) + C$

Problem 2

Compute $\displaystyle \int x^{15}(x^{16} + 5)^{5/2}\ dx $

• Hint 1

  What should an inside function be for this?

• Hint 2

  Try $u = x^{16} + 5$. How does $du$ factor into this integral?

• Answer

  $\frac{1}{56}(x^{16} + 5)^{7/2} + C$

Problem 3

Compute $\displaystyle \int x^5(x^2 - 3)^{3/2}\ dx $

• Hint 1

  What should an inside function be for this? Try making that $u$.

• Hint 2

  After the substitution, you end up with both x and u in the expression. You need to make that only in terms of $u$. How can you do this? How can you get rid of $x$?

• Hint 3

  Try $u = x^2 - 3$. This means that $x^2 = u+3$, so that $x^4 = (u+3)^2$.

• Hint 4

  The integral reduces to $\displaystyle \int \frac{1}{2}(u+3)^2u^{3/2} du$

• Answer

  $\frac{1}{315}(x^2 - 3)^{5/2}(35x^4 + 60x^2 + 72) + C$

Problem 4

Compute $\displaystyle \int_3^6 3x(x^2 +4)^3\ dx$

• Hint 1

  What should an inside function be for this?

• Hint 2

  Try $u = x^2 + 4$. How does $du$ factor into this integral?

• Hint 3

  Don't forget to change the bounds on the integral.

• Hint 4

  The integral reduces to $\displaystyle \int_{13}^{40} \frac{3}{2}u^3\ du$.

• Answer

  $949290$

Problem 5

Compute $\displaystyle \int_{0}^4 \frac{2x^2}{x^3 + 2}\ dx$

• Hint 1

  What should we treat as an inside function here?

• Hint 2

  Try $u = x^3 + 2$. What is $du$?

• Hint 3

  The integral should reduce to $\displaystyle \int_2^{66} \frac{2}{3} \frac{1}{u} \ du$

• Answer

  $\frac{2}{3} \ln{33}$ (after applying logarithm rules)

Problem 6

Compute $\displaystyle \int \frac{x^2}{x^6 + 5}\ dx$

• Hint 1

  What is your first thought for $u$? Does this work for this integral?

• Hint 2

  If you try $u = x^6 + 5$, there's no way to make the $du$ work. If this doesn't work, a last option would be the inverse trig integrals. Does one of these apply here?

• Hint 3

  Try to make this look like the derivative of inverse tangent. You'll need to take care of the $x^6$ term as well as the $5$.

• Hint 4

  Write the integrand as $\frac{1}{5} \frac{x^2}{(\frac{x^6}{5} + 1 )}$ and the set $u = \frac{x^3}{\sqrt{5}}$.

• Answer

  $\frac{1}{3\sqrt{5}} \tan^{-1}(\frac{x^3}{\sqrt{5}}) + C$

Submission Problems

Problem 1

Compute $\displaystyle \int \frac{x^3 + x}{(x^4 +2x^2 + 3)^3}\ dx $

Problem 2

Compute $\displaystyle \int_1^2 e^x \sin(e^{x} + 4)\ dx$