Exam 9: Techniques of Integration

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 2 x ^ { 3 } + 5 x ^ { 2 } + 14 x + 7 } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } } d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { d x } { x ^ { 2 } \left( x ^ { 2 } - 25 \right) }$

Solve the problem. -The pulse rate for a species of small mammal follows a normal distribution with a mean $\mu = 205$ beats per minute and standard deviation $\sigma = 30$ beats per minute. In a population of 800 of these mammals, how many would you expect to have a pulse rate between 193 and 241 ?

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { 0 } 7 x e ^ { 3 x } d x$

Find the area or volume. -Find the area of the region in the first quadrant between the curve $y = e ^ { - 6 x }$ and the $x$ -axis.

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { \infty } 22 x e ^ { - x } d x$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \frac { \sqrt { x } } { \sqrt { 9 - x } } d x$

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - $f ( x ) = \frac { k } { x } \text { over } [ 1,17 ]$

Evaluate the integral. - $\int \frac { ( x + 7 ) ^ { 2 } \tan ^ { - 1 } x + ( 4 x - 22 ) ( x + 7 ) } { \left( x ^ { 2 } + 1 \right) ( x + 7 ) ^ { 2 } } d x$

Evaluate the integral. - $\int _ { \pi / 12 } ^ { \pi / 2 } ( 1 - \cos 4 x ) \cos 2 x d x$

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

Expand the quotient by partial fractions. - $\frac { x ^ { 2 } - 4 x + 28 } { x ^ { 2 } - 10 x + 24 }$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 60 } { t ^ { 3 } + 4 t ^ { 2 } - 5 t } d t$

Solve the problem by integration. -The slope of a curve is given by $\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 87 \mathrm { x } ^ { 2 } + 128 } { 9 x ^ { 4 } + 16 \mathrm { x } ^ { 2 } }$ . Find the equation of the curve if it passes through $\left( \frac { 4 } { 3 } , 4 \right)$ .

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int _ { 1 } ^ { 5 } \frac { 3 x ^ { 2 } + x + 50 } { \left( x ^ { 2 } + 25 \right) ( x + 2 ) } d x$

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 1 } \frac { 9 } { 1 + x } d x$

Solve the initial value problem for y as a function of x. - $\left( x ^ { 2 } + 49 \right) \frac { d y } { d x } = 1 , y ( 7 ) = 0$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 72 d x } { x ^ { 3 } - 9 x }$

Evaluate the integral. The integral may not require integration by parts. - $\int \frac { \ln x } { x ^ { 3 } } d x$

Provide an appropriate response. - $\text { A student knows that } \int _ { a } ^ { + \infty } \mathrm { dx } = 57 . \text { Can } \int _ { - \infty } ^ { - 1 } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text { be found, and if so, what is it? }$