Exam 8: Techniques of Integration

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { x }$ , and the line $x = \ln 2$ about the line $x = \ln 2$ .

Integrate the function. - $\int \frac { 40 d x } { x ^ { 2 } \sqrt { x ^ { 2 } + 64 } }$

Integrate the function. - $\int \sqrt { 36 - x ^ { 2 } } d x$

Use integration by parts to establish a reduction formula for the integral. - $\int x ^ { n } e ^ { x } d x$

Integrate the function. - $\int \frac { \sqrt { x ^ { 2 } + 9 } } { 5 x ^ { 2 } } d x$

Evaluate the integral. - $\int \sin 7 x \cos 4 x d x$

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$ -axis and the curve $y = x \cos x , 0 \leq x \leq \pi / 2$ about the $y$ -axis.

Use integration by parts to establish a reduction formula for the integral. - $\int \sin ^ { n } x d x$

Evaluate the integral. -Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Evaluate the integral. -Use the formula $\int \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \mathrm { dx } = \mathrm { xf } ^ { - 1 } ( \mathrm { x } ) - \int \mathrm { x } \left( \frac { \mathrm { d } } { \mathrm { dx } } \mathrm { f } ^ { - 1 } ( \mathrm { x } ) \right) \mathrm { dx }$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Solve the problem. -The rate of water usage for a business, in gallons per hour, is given by $W ( t ) = 16 t e ^ { - t }$ , where $t$ is the number of hours since midnight. Find the average rate of water usage over the interval $0 \leq t \leq 5$ .

Evaluate the integral. - $\int _ { \pi / 24 } ^ { \pi / 12 } \cot ^ { 4 } 6 t d t$

Evaluate the integral. - $\int \cos 7 x \cos 3 x d x$

Use integration by parts to establish a reduction formula for the integral. - $\int x ^ { n } e ^ { - x ^ { 2 } } d x$

Evaluate the integral. - $\int _ { \pi / 3 } ^ { \pi / 2 } \sqrt { 1 + \cos x } d x$

Evaluate the integral. - $\int \sec ^ { 3 } 9 x d x$

Evaluate the integral. - $\int _ { 0 } ^ { 1 / 2 } 3 \sin ^ { 4 } 2 \pi x d x$

Use any method to evaluate the integral. - $\int \frac { \tan ^ { 2 } x } { \csc x } d x$

Evaluate the integral. - $\int _ { - \pi / 2 } ^ { \pi / 2 } \cos ^ { 5 } 5 x d x$

Find the area between $y = \ln x$ and the $x$ -axis from $x = 1$ to $x = 5$ .