Exam 9: Techniques of Integration

Determine whether the function is a probability density function over the given interval. - $f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { \pi \left( x ^ { 2 } + 1 \right) } & x \geq 0 \\0 & x < 0\end{array} \right.$

Determine whether the function is a probability density function over the given interval. - $f ( x ) = 9 ^ { x } \text { over } \left[ 0 , \frac { \ln ( 1 + \ln 9 ) } { \ln 9 } \right]$

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

Evaluate the integral. - $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln 6 x d x$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \frac { \cos \theta } { \sin \theta \sqrt { 49 + \sin ^ { 2 } \theta } } d \theta$

Find the indicated probability. - $\mathrm { f } ( \mathrm { x } ) = \mathrm { e } ^ { - \mathrm { x } ; } ; [ 0 , \infty ) , \mathrm { P } ( \mathrm { x } \geq 7 )$

Evaluate the integral. - $\int \cos ^ { - 1 } x d x$

Solve the problem. -The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 5.0 minutes and a standard deviation of 1 minute. Find the probability that a Randomly selected college student will take between 3.5 and 6.0 minutes to find a parking spot in the library lot.

Evaluate the integral. - $\int \sin 4 t \sin \frac { t } { 3 } d t$

Provide an appropriate response. -(a) Express $\frac { 3 x - 1 } { x ^ { 2 } - 3 x - 10 }$ as a sum of partial fractions. (b) Evaluate $\int \frac { 3 x - 1 } { x ^ { 2 } - 3 x - 10 } d x$ . (c) Evaluate $\int \frac { 2 x ^ { 2 } - 3 x - 21 } { x ^ { 2 } - 3 x - 10 } d x$ . (d) Find a solution to the initial value problem: $\frac { d y } { d x } = \frac { 3 x y - y } { x ^ { 2 } - 3 x - 10 } , \quad y ( 4 ) = 12 .$

Evaluate the integral by making a substitution and then using a table of integrals. - $\int \tan ^ { - 1 } \sqrt { x + 2 } d x$

Solve the problem. -Find the area between y = ln x and the x-axis from x = 1 to x = 3.

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { x ^ { 3 } } { x ^ { 2 } + 2 x + 1 } d x$

Determine whether the improper integral converges or diverges. - $\int _ { - \infty } ^ { - 3 } \left( e ^ { 100 x } + x ^ { 3 } \right) d x$

Evaluate the integral. - $\int \frac { d x } { x \sqrt { 49 x ^ { 2 } - 16 } }$

Solve the problem. -Find the average value of the function $y = \frac { 16 } { \sqrt { 9 - 16 x ^ { 2 } } }$ over the interval from $x = 0$ to $x = \frac { 3 } { 8 }$ .

Solve the problem. -Find the area bounded by $y \left( 36 + 4 x ^ { 2 } \right) = 2 , x = 0 , y = 0$ , and $x = 4$ .

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. $\int \sin ^ { - 1 } x d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { 0 } 4 e ^ { x } \sin x d x$

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - $\int e ^ { t } \sec ^ { 3 } \left( e ^ { t } - 2 \right) d t$