Exam 9: Techniques of Integration

Provide an appropriate response. -A student wishes to find the integral $\int _ { 0 } ^ { + \infty } \mathrm { f } ( \mathrm { x } ) \mathrm { dx }$ of a function that has the property limit $\lim _ { \mathrm { x } \rightarrow \infty } \mathrm { f } ( \mathrm { x } ) = 1$ . Why can this not be done?

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$ -axis and the curve $y = x \cos x , 0 \leq x \leq \pi / 2$ about the $y$ -axis.

Use integration by parts to establish a reduction formula for the integral. - $\int \csc ^ { n } x d x , n \neq 1$

Evaluate the integral. - $\int \tan ^ { 5 } 4 x d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { e } 11 e ^ { - x } d x$

Integrate the function. - $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 25 } } , x > 5$

Solve the problem. -Find the length of the curve $y = \ln ( \sin x ) , \pi / 3 \leq x \leq \pi / 2$

Evaluate the integral. - $\int 14 x \sin x d x$

Evaluate the integral. - $\int _ { 2 } ^ { 4 } 3 x \ln x d x$

Solve the problem. -Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 1 } ^ { 2 } \left( 4 x ^ { 5 } - 2 x \right) d x$ with $n = 6$ steps.

Evaluate the integral. - $\int \mathrm { e } ^ { 2 x } \mathrm { x } ^ { 2 } \mathrm { dx }$

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

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

Evaluate the integral. - $\int \frac { 3 e ^ { 2 t } + 4 e ^ { t } } { e ^ { 2 t } - 10 e ^ { t } + 25 } d t$

Evaluate the integral. - $\int \frac { 4 e ^ { 2 t } - 7 e ^ { t } } { e ^ { 3 t } - 3 e ^ { 2 t } + e ^ { t } - 3 } d t$

Solve the problem. -The length of the ellipse $x = a \cos t , y = b \sin t , 0 \leq t \leq 2 \pi$ is $\text { Length } = 4 \mathrm { a } \int _ { 0 } ^ { \pi / 2 } \sqrt { 1 - \mathrm { e } ^ { 2 } \cos ^ { 2 } \mathrm { t } } \mathrm { dt }$ where $\mathrm { e }$ is the ellipse's eccentricity. Use Simpson's Rule with $n = 6$ to estimate the length of the ellipse when $a = 2$ and $e = \frac { 1 } { 3 }$ .

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { e ^ { x } } { \sqrt { 1 + x ^ { 2 } } } d x$

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { 6 y ^ { 4 } + 3 y ^ { 2 } - 6 y } { y ^ { 3 } - 1 } d y$

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { \ln | x | } { x } d x$

Use any method to evaluate the integral. - $\int \frac { \tan ^ { 2 } x } { \csc x } d x$