Exam 9: Techniques of Integration

$\int \mathrm { e } ^ { 2 x } \cos \left( \mathrm { e } ^ { 2 \mathrm { x } } \right)$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

Approximate $\int _ { 0 } ^ { 4 }$ $\frac { 1 } { x ^ { 2 } + 1 }$ dx; n = 2, by (a) the midpoint rule, (b) the trapezoidal rule, and (c) Simpson's rule. Enter your answers in that order as just unlabeled real numbers rounded to two decimal places, separated by commas.

Determine the integral by making an appropriate substitution. - $\int ( x - 1 ) ^ { 7 } d x$

Evaluate the integral using integration by parts. - $\int e ^ { a x } \cos b x d x$

Evaluate the integral using integration by parts. - $\int x ^ { 2 } e ^ { - x } d x$

Determine the integral by making an appropriate substitution. - $\int ( \cos x ) ^ { 8 } \sin x d x$

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

Determine the integral by making an appropriate substitution. - $\int 3 ( 2 x + 5 ) ^ { 3 } d x$

Decide whether integration by parts or substitution should be used to compute the indefinite integral $\int$ $\frac { - \mathrm { e } ^ { 1 / \mathrm { x } } } { \mathrm { x } ^ { 2 } }$ dx If substitution, indicate the value of u; if by parts, indicate f(x) and g(x).

Which of the following statements are true?

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

Approximate $\int _ { 0 } ^ { 4 } \sqrt { x } d x$ ; n = 2, 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, b are real numbers rounded to two decimal places, and c is a reduced fraction of form $\frac { a } { b }$ all unlabeled and answering the questions in order, separated by commas.

Evaluate the integral. - $\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { x ^ { 2 } } \mathrm { dx }$

Does this integral $\int _ { 0 } ^ { 1 } x e ^ { 3 x ^ { 2 } }$ dx = $\frac { 1 } { 6 }$ ( $\mathrm { e } ^ { 3 }$ - 1)?

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

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

$\int$ $\frac { x ^ { 2 } + 2 } { \sqrt { x ^ { 3 } + 6 x } }$ dx Enter your answer with any coefficients in front as integers or reduced fractions of form $\frac { a } { b }$ .

$\int ^ { * }$ $\frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$ dx Enter your answer as a reduced fraction or the word "divergent".

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

Does this integral $\int x e ^ { - 3 x } \mathrm { dx }$ = - $\frac { 1 } { 3 }$ $x e ^ { - 3 x }$ - $\frac { 1 } { 9 }$ $e ^ { - 3 x }$ + C?