Exam 8: Techniques of Integration

The integral $\int \sqrt { 4 x ^ { 2 } + 9 } d x$ is

The integral $\int \frac { x ^ { 3 } + 2 x ^ { 2 } - 1 } { x ^ { 2 } - 1 } d x$ is

The improper integral $\int _ { 1 } ^ { 3 } \frac { x } { \sqrt { 3 - x } } d x$ is

Suppose you want to use the Midpoint Rule with n = 4 and equal-length subintervals to approximate $\int _ { 1 } ^ { 6 } e ^ { - x } d x$ ? What are the sample points you would use to compute the heights of the rectangles used in this approximation

The integral $\int x e ^ { - 2 x } d x$ is

The number Cof chipmunks in a small rural township t days after May 1, 2015, follows the logistic model $\frac { d C } { d t } = 1.56 C \left( 1 - \frac { C } { 2,000 } \right)$ . The number of chipmunks recorded on May 1, 2015, was 775. What is the carrying capacity in this scenario?

The integral $\int \tan ^ { 2 } x \sec ^ { 2 } x d x$ is

Suppose you want to use the Midpoint Rule with n = 4 and equal-length subintervals to approximate $\int _ { 1 } ^ { 6 } e ^ { - x } d x$ ? What is the approximation using the Midpoint Rule?

The integral $\int \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$ is

The integral $\int \sqrt { 16 x ^ { 2 } + 9 } d x$ is

Using Simpson's Rule with n = 8, the approximated value of $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } d x,$ to three decimal places, is

The integral $\int \log _ { \frac { 1 } { 2 } } x d x$ is

The integral $\int x \cos x d x$ is

The integral $\int x 2 ^ { x } d x$ is

The integral $\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 4 } } d x$ is

The integral $\int \frac { x + 3 } { \sqrt { x ^ { 2 } - 6 x + 10 } } d x$ is

The integral $\int x ^ { 2 } \sqrt { 9 x ^ { 2 } + 1 } d x$ is

The integral $\int \sin ^ { 3 } x \cos ^ { 3 } x d x$ is

The integral $\int e ^ { 2 x } \cos x d x$ is

Using the Trapezoidal Rule with n = 4, the approximated value of $\int _ { 0 } ^ { 2 } e ^ { - \frac { x ^ { 2 } } { 2 } } d x,$ to three decimal places, is