Exam 9: Techniques of Integration

Does this integral $\int x \sqrt { x + 3 } d x$ = $\frac { 2 } { 3 }$ x $( x + 3 ) ^ { 3 / 2 }$ - $\frac { 4 } { 15 }$ $( x + 3 ) ^ { 5 / 2 }$ + C?

$\int \cot x \csc x e ^ { \csc x }$ dx   [Hint: $\frac { \mathrm { d } } { \mathrm { dx } }$ csc x = -cot x csc x] 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 integral. - $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 5 + 3 x ^ { 3 } \right) ^ { 2 } }$ dx

Evaluate the integral. - $\int _ { 2 } ^ { 3 } \frac { x } { \left( x ^ { 2 } - 2 \right) ^ { 2 } } d x$ Enter just a reduced fraction of form $\frac { a } { b }$ .

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

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

Determine the integral by making an appropriate substitution. - $\int \tan ^ { 9 } ( 2 x ) \sec ^ { 2 } ( 2 x )$ dx

Approximate $\int _ { 0 } ^ { 3 } x ^ { 2 } d x$ ; n = 4, by (a) the trapezoidal rule, (b) the midpoint rule, and (c) then find the exact value of the integral. Enter just a, b, c where a and b are real numbers to two decimal places (rounded off), and c is an integer, all separated by commas and answering (a), (b), (c) in order but unlabeled.

$\int _ { 0 } ^ { \infty } \ln x ^ { 2 } d x$ Enter your answer as just a reduced fraction or the word "divergent".

Approximate $\int _ { 0 } ^ { 4 } 2 x d x$ ; n = 4, by (a) the trapezoidal rule, (b) the midpoint rule, and (c) then find the exact value of the integral. Enter just three integers separated by commas answering (a), (b), (c) in that order but unlabeled.

$\int _ { 0 } ^ { x }$ $\frac { 1 } { \sqrt { x + 1 } }$ dx Enter your answer as a reduced fraction or the word "divergent".

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

Decide whether integration by parts or substitution should be used to compute the indefinite integral $\int$ $\frac { \cos ( \ln x ) } { x }$ dx If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

Evaluate the integral using integration by parts. - $\int x$ $\sqrt { x + 1 }$ dx

Evaluate the integral using integration by parts. - $\int ( x + 1 ) e ^ { x } d x$

Does this integral $\int x \ln x d x$ = $\frac { x ^ { 2 } } { 2 }$ ln x - $\frac { x ^ { 2 } } { 4 }$ + C?

Evaluate the integral using integration by parts. - $\int x \mathrm { e } ^ { - x } \mathrm { dx }$

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

$\int x \sqrt { x ^ { 2 } - 4 } d x$ 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. - $\int \frac { x } { \left( 7 x ^ { 2 } + 3 \right) ^ { 5 } }$ dx