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:
Review u substitution

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:
Again, we need to use substitution. Let $u=\sin(x)$. Then, $du=\cos(x)dx$. Also, we need to change the bounds of integration.
Plugging in our values into the equation $u=\sin(x)$, we get $u_{1}=\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}={\frac {\sqrt {2}}{2}}$ and $u_{2}=\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}=1$.
Therefore, the integral becomes $\int _{\frac {\sqrt {2}}{2}}^{1}{\frac {1}{u^{2}}}~du$.

Step 2:
We now have:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int _{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos(x)}{\sin^2(x)}~dx=\int_{\frac{\sqrt{2}}{2}}^1 \frac{1}{u^2}~du=\left.\frac{-1}{u}\right|_{\frac{\sqrt{2}}{2}}^1=-\frac{1}{1}-\frac{-1}{\frac{\sqrt{2}}{2}}=-1+\sqrt{2}} .

Final Answer:
(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{9}(1+x^3)^{\frac{3}{2}}+C}
(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1+\sqrt{2}}

Return to Sample Exam