Exercise 15

For each of the following indefinite integrals, determine whether you would use $u$-substitution, integration by parts, neither*, or both to evaluate the integral. In each case, write one sentence to explain your reasoning, and include a statement of any substitutions used. (That is, if you decide in a problem to let $u = e^{3x}$, you should state that, as well as that $du = 3e^{3x} \, dx$.) Finally, use your chosen approach to evaluate each integral. (* one of the following problems does not have an elementary antiderivative and you are not expected to actually evaluate this integral; this will correspond with a choice of “neither” among those given.)

1. $\displaystyle \int x^2 \cos(x^3) \, dx$

2. $\displaystyle \int x^5 \cos(x^3) \, dx$  (Hint: $x^5 = x^2 \cdot x^3$)

3. $\displaystyle \int x\ln(x^2) \, dx$

4. $\displaystyle \int \sin(x^4) \, dx$

5. $\displaystyle \int x^3 \sin(x^4) \, dx$

6. $\displaystyle \int x^7 \sin(x^4) \, dx$