Exam 9: Techniques of Integration

Solve the problem. -The number of new mini-vans sold by a particular salesperson during the month of March is exponentially distributed with a mean of 6. What is the probability that the salesperson will sell between 2 and 4 mini-vans in March?

Evaluate the improper integral or state that it is divergent. - $\int _ { 1 } ^ { \infty } \frac { 14 } { 8 x ( x + 1 ) ^ { 2 } } d x$

Evaluate the integral. - $\int ( 2 x - 1 ) \ln ( 4 x ) d x$

Evaluate the integral. - $\int x \csc ^ { 2 } 3 x d x$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 3 } ^ { 6 } \frac { 1 } { ( x - 1 ) ^ { 2 } } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { 5 x + 7 } { 8 x ^ { 3 } + 6 x ^ { 2 } + 1 }$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 6 x - 12 } { x ^ { 2 } - 4 x - 5 } d x$

Evaluate the integral. - $\int \frac { \sqrt { 4 - x ^ { 2 } } } { x } d x$

Provide an appropriate response. -i) Show that $\int _ { 0 } ^ { \infty } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x$ diverges, and hence $\int _ { - \infty } ^ { \infty } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x$ diverges. ii) Show that $\lim _ { b \rightarrow \infty } \int _ { - b } ^ { b } \frac { 4 x ^ { 3 } } { x ^ { 4 } + 1 } d x = 0$ .

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 2 } 8 x ^ { 2 } d x$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \frac { e ^ { 2 x } } { 2 e ^ { x } + 3 } d x$

Evaluate the integral. - $\int \frac { - \sin t ( 3 \cos t + 4 ) d t } { \cos ^ { 2 } t - 4 \cos t + 4 }$

Provide an appropriate response. -The standard normal probability density function is defined by $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } / 2 }$ . (a) Show that $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } x \mathrm { e } ^ { - x ^ { 2 } / 2 } \mathrm { dx } = \frac { 1 } { \sqrt { 2 \pi } }$ . (b) Use the result in part (a) to show that the standard normal probability density function has mean 0 .

Evaluate the integral. - $\int \sin 4 x \cos 2 x d x$

Solve the initial value problem for y as a function of x. - $\sqrt { 49 - x ^ { 2 } } \frac { d y } { d x } = 1 , x < 7 , y ( 0 ) = 12$

Find the indicated probability. - $f ( x ) = 3 x \left( \frac { 4 } { 3 } - x \right) \text { over } [ 0,1 ] , P ( 0.8 < x )$

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

Evaluate the integral by using a substitution prior to integration by parts. - $\int \frac { 1 } { 2 } \mathrm { e }^{ \sqrt { 7 x + 8 }}\mathrm { dx }$

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { 0 } \frac { 15 } { ( x - 1 ) ^ { 2 } } d x$

Determine whether the improper integral converges or diverges. - $\int _ { 0 } ^ { \pi / 2 } \frac { \sin \sqrt { t } } { \sqrt { t } } d t$