009B Sample Midterm 2

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar. Click on the

 boxed problem numbers  to go to a solution.

 Problem 1 

This problem has three parts:

a) State the Fundamental Theorem of Calculus.

b) Compute   ${\frac {d}{dx}}\int _{0}^{\cos(x)}\sin(t)~dt$.

c) Evaluate $\int _{0}^{\pi /4}\sec ^{2}x~dx$.

 Problem 2 

Evaluate

a) $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

b) $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

 Problem 3 

Evaluate

a) $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

b) $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

 Problem 4 

Evaluate the integral:

$\int e^{-2x}\sin(2x)~dx$

 Problem 5 

Evaluate the integral:

$\int \tan ^{4}x~dx$