Exam 7: Techniques of Integration

A body moves along a coordinate line in such a way that its velocity at any time $t$ , where $0 \leq t \leq 6$ , is given by $v ( t ) = t \sqrt { 36 - t ^ { 2 } } .$ Find its position function if it is initially located at the origin.

Determine whether the improper integral converges or diverges, and if it converges, find its value. $\int _ { - 27 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } d x$

Evaluate the integral. $\int _ { x / 6 } ^ { x / 3 } \frac { 3 \ln ( \tan x ) } { 7 \sin x \cos x } d x$

Use a table of integrals to evaluate the integral. Select the correct answer. $\int x \sqrt { 3 + 3 x } d x$

Evaluate the integral. Select the correct answer. $\int _ { 0 } ^ { 8 } \left( x ^ { 2 } + 8 \right) e ^ { - x } d x$

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Evaluate the integral. $\int e ^ { 5 x } \cos 8 x d x$

Find the integral. $\int \frac { d x } { x ( x - 3 ) }$

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 0 } ^ { \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { 4 - x ^ { 2 } } } d x$

Find the integral using an appropriate trigonometric substitution. $\int \frac { \sqrt { 25 x ^ { 2 } - 16 } } { x } d x$

Find the integral using an appropriate trigonometric substitution. $\int x \sqrt { 1 - x ^ { 2 } } d x$

Find the integral. $\int x ^ { 5 } \ln x d x$

Find the integral using an appropriate trigonometric substitution. Select the correct answer. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 1 } } d x$

Evaluate the integral. $\int x \sin 3 x \cos 3 x d x$

The region under the curve $y = 6 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$ -axis. Find the volume of the resulting solid.

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Evaluate the integral. $\int 4 y ^ { 3 } e ^ { y } d y$

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

Find the integral. $\int \frac { 3 x - 9 } { x ^ { 2 } - 4 x - 5 } d x$

Evaluate the integral. $\int \frac { \cos ( \ln 18 t ) } { t } d t$