009B Sample Midterm 3, Problem 3

Compute the following integrals:

a) $\int x^{2}\sin(x^{3})dx$

b) $\int _{-{\frac {\pi }{4}}}^{\frac {\pi }{4}}\cos ^{2}(x)\sin(x)dx$

Foundations:
u substitution

Solution:

(a)

Step 1:
We proceed using u substitution. Let $u=x^{3}$. Then, $du=3x^{2}dx$.
Therefore, we have
$\int x^{2}\sin(x^{3})dx=\int {\frac {\sin(u)}{3}}du$

Step 2:
We integrate to get
$\int x^{2}\sin(x^{3})dx={\frac {-1}{3}}\cos(u)+C={\frac {-1}{3}}\cos(x^{3})+C$

(b)

Step 1:

Step 2:

Final Answer:
(a) ${\frac {-1}{3}}\cos(x^{3})+C$
(b)

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