Good evening all! Since test #1 is coming up on next Thursday (10/8/2020), it is important to review all of the integration techniques we have learned so far. Integration by substitution is one of them!

I have decided to do problems #9 and #14 in the Integration by Substitution homework set in WeBWorK. This post will be updated over time based on feedback I receive.

Problem #9

  1. This is a problem of moderate difficulty. You have to evaluate the indefinite integral.
  2. Find u: (u =x6 + 5; du = x5 dx).
  3. Perform u-substitution. Note that x5 has been divided, so it has been turned to 1/6 according to the power rule.
  4. Use the natural log rule.
  5. Put the original value (x6 + 5) back in place of u.

Problem #14

  1. This is a problem of harder difficulty, with multiple steps. You have to evaluate the definite integral, and find out if the integral is defined or not.
  2. Find u: (u = 3x – 4; du = 3 dx). You should divide 3 by both sides in order for du to fit the form of dx.
  3. Perform u-substitution.
  4. Use the constant multiple rule for 1/3 and 5. (Hint: 5 × 1/3)
  5. Use the power rule for u-23.
  6. Simplify by multiplying.
  7. Put the original value (3x-4) back in place of u.
  8. Evaluate the integral. Remember that the upper bounds are 8/3 (or 2 ⅔), and lower bounds are 0. Then, simplify accordingly based on the values you substituted in their respective places.

Note: Since 422 is an extremely large number, I have to round this number to three significant figures (1.76 × 1013), and use scientific notation in this case.

With enough simplification, you will notice that the denominators are alike, and you can now combine -5 and +5.

Therefore, the definite integral is 0.

Motivating questions

  1. In problem #9, in Step 3, how did I turn x5 to 1/6 before performing u-substitution?
  2. In problem #14, WeBWorK’s answer is “The integral does not exist”, but I wrote that the integral is 0. What are the possible errors I have made in my step-by-step procedure?