009B Sample Final 3, Problem 2

Evaluate the following integrals.

(a)  $\int _{0}^{\frac {\sqrt {3}}{4}}{\frac {1}{1+16x^{2}}}~dx$

(b)  $\int {\frac {x^{2}}{(1+x^{3})^{2}}}$

(c)  $\int _{1}^{e}{\frac {\cos(\ln(x))}{x}}~dx$

Foundations:
$\int {\frac {1}{1+x^{2}}}~dx=\arctan(x)+C$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)

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