009B Sample Midterm 1, Problem 1

Evaluate the indefinite and definite integrals.

a) $\int x^{2}{\sqrt {1+x^{3}}}dx$

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

Foundations:

Solution:

(a)

Step 1:
We need to use substitution. Let $u=1+x^{3}$. Then, $du=3x^{2}dx$ and ${\frac {du}{3}}=x^{2}dx$.
Therefore, the integral becomes ${\frac {1}{3}}\int {\sqrt {u}}du$.

Step 2:
We now have:
$\int x^{2}{\sqrt {1+x^{3}}}dx={\frac {1}{3}}\int {\sqrt {u}}du={\frac {2}{9}}u^{\frac {3}{2}}+C={\frac {2}{9}}(1+x^{3})^{\frac {3}{2}}+C$.

(b)

Step 1:

Step 2:

Final Answer:
(a) ${\frac {2}{9}}(1+x^{3})^{\frac {3}{2}}+C$
(b)

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