Exam 9: Techniques of Integration

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 0 } ^ { 5 } \sqrt { x + 4 } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Determine whether the improper integral converges or diverges. - $\int _ { 0 } ^ { \pi } \frac { \sin \theta d \theta } { ( \pi - \theta ) ^ { 2 / 5 } }$

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - $f ( x ) = \frac { 1 } { 7 } x \text { over } [ 4 , k ]$

Evaluate the integral. - $\int \frac { d x } { 5 + 13 \sin 2 x }$

Use reduction formulas to evaluate the integral. - $\int \tan ^ { 5 } 2 x d x$

Solve the problem by integration. -Find the first-quadrant area bounded by $y = \frac { 1 } { x ^ { 3 } + 4 x ^ { 2 } + 3 x } , x = 1$ , and $x = 4$ .

Evaluate the integral. - $\int \tan ^ { 4 } 2 t d t$

Evaluate the integral. - $\int \csc ^ { 3 } 6 t d t$

Integrate the function. - $\int \frac { x ^ { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 5 / 2 } } d t$

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

Solve the problem. -Find the volume generated by revolving the curve $y = \cos 5 x$ about the $x$ -axis, $0 \leq x \leq \pi / 12$

Use any method to evaluate the integral. - $\int 5 x \sin ^ { 2 } x d x$

Solve the initial value problem for x as a function of t. - $\left( t ^ { 2 } + 2 t \right) \frac { d x } { d t } = 2 x + 8 , x ( 1 ) = 1$

Evaluate the integral. - $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 7 } y d y$

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

Evaluate the integral. - $\int \sin 4 x \cos 2 x d x$

Solve the problem. -Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 0 } ^ { 5 } ( 4 \mathrm { x } + 1 ) \mathrm { dx }$ with $\mathrm { n } = 6$ steps.

Evaluate the integral. - $\int \frac { \sqrt { 2 x - 7 } } { x ^ { 2 } } d x$

Determine whether the improper integral converges or diverges. - $\int _ { - \infty } ^ { \infty } 5 x e ^ { - x ^ { 2 } } d x$

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 1 } \frac { 9 } { 1 + x ^ { 2 } } d x$