Exam 9: Techniques of Integration

$\int$ $\frac { \cos x - \sin x } { \cos x + \sin x }$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

$\int$ $\frac { \sin x e ^ { ( x + \cos x ) } } { e ^ { x } }$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ . No parentheses around arguments of functions.

Evaluate the improper integral whenever it is convergent. If it is divergent, state this. - $\int _ { 2 } ^ { \infty } \frac { d x } { x ( \ln x ) ^ { 2 } }$

$\int x \left( x ^ { 2 } + 1 \right) ^ { 17 } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Does this integral $\int x \cos x d x$ = x sin x + cos x + C?

Does this integral $\int \sin x \sec ^ { 2 } x d x$ = sin x tan x + cos x + C?

Determine the integral by making an appropriate substitution. - $\int \frac { e ^ { 1 / x } } { x ^ { 2 } }$ dx

$\int$ $\frac { \tan x d x } { \ln ( \cos x ) }$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Determine the integral by making an appropriate substitution. - $\int x ^ { 2 } \sec ^ { 2 } x ^ { 3 }$ dx

Determine the integral by making an appropriate substitution. - $\int \frac { 9 x ^ { 2 } - 2 } { \left( 3 x ^ { 3 } - 2 x \right) ^ { 2 } }$ dx

Does this integral $\int x ( x + 2 ) ^ { 2 / 3 } d x$ + $\frac { 3 } { 40 }$ $( 2 + x ) ^ { 5 / 3 }$ (5x - 6) + C?

Does this integral $\int \left( x ^ { 2 } + x \right) e ^ { x } d x$ = $e ^ { x }$ ( $x ^ { 2 }$ + x - 1) + C?

Approximate $\int _ { 0 } ^ { 1 } x ^ { 4 }$ dx; n = 6, by (a) Simpson's rule and (b) the trapezoidal rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by a comma.

$\int \left( x ^ { 3 } - 3 x ^ { 2 } \right) \left( 6 x ^ { 2 } - 12 x \right) d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form a/b.

Recall that the capital value of an asset is given by $\int _ { 0 } ^ { \infty } \mathrm { K } ( \mathrm { t } ) \mathrm { e } ^ { - r \mathrm { t } } \mathrm { dt }$ If the annual rate of earnings, K(t), of a certain machine is $3000 and if the interest rate is 12%, find the capital interest of the machine. Enter just an integer representing dollars (no units).

Determine the integral by making an appropriate substitution. - $\int \frac { 4 x } { \sqrt { x - 1 } }$ dx

$\int x e ^ { 3 x ^ { 2 } } d x$ Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int _ { 8 / 3 } ^ { \infty } ( 3 x + 1 ) ^ { - 3 / 2 } \mathrm { dx }$ Enter your answer as just a reduced fraction or the word "divergent".

Approximate $\int _ { 1 } ^ { 3 }$ $\frac { 1 } { x }$ dx; n = 4 Enter just a real number rounded to two decimal places.

Determine the integral by making an appropriate substitution. - $\int$ $\frac { \ln 5 x } { x }$ dx   Use the substitution u = ln 5x.