Exam 7: Techniques of Integration

Determine whether the improper integral converges or diverges, and if it converges, find its value, $\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 3 } } d x$ Select the correct answer.

Evaluate the integral. $\int _ { - 1 } ^ { 1 / \sqrt { 3 } } \frac { e ^ { \operatorname { sintan } y } } { 1 + y ^ { 2 } } d y$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } + 6 x ^ { 2 } + 7 x + 2 } { ( x + 1 ) ^ { 2 } \left( x ^ { 2 } + 1 \right) } d x$

Evaluate the integral or show that it is divergent. $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 6 } } d x$

A manufacturer of light bulbs wants to produce bulbs that last about 600 hours but, of course, some bulbs burn out faster than others. Let $F ( t )$ be the fraction of the company's bulbs that burn out before $t$ hours. $F ( t )$ lies between 0 and 1 . Let $r ( t ) = F ^ { \prime } ( t )$ . What is the value of $\int _ { 0 } ^ { \infty } r ( t ) d t$ ?

Evaluate the integral. $\int \frac { \cos x } { 25 + \sin ^ { 2 } x } d x$

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

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

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

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 36 } d x$

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

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 49 } } ; x = 7 \sec \theta$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

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

Find the integral. Select the correct answer. $\int \tan ^ { 3 } x \sec ^ { 5 } x d x$

Find the integral. $\int x e ^ { 3 x } d x$

Evaluate the integral or show that it is divergent. Select the correct answer. $\int _ { - \infty } ^ { \infty } \frac { 5 d x } { 4 x ^ { 2 } + 4 x + 5 }$

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

Find the integral. $\int \cos ^ { 5 } x \sin ^ { 2 } x d x$