Exam 13: Techniques of Integration

Find the capitalized cost C of an asset forever. The capitalized cost is given by $C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - r t } d t$ where $C _ { 0 } = \$ 500,000$ is the original investment, t is the time in years, r = 12% is the annual interest rate compounded continuously, n is the total time in years over which the asset is capitalized, and $c ( t ) = 25,000 ( 1 + 0.08 t )$ is the annual cost of maintenance (measured in dollars). Round your answer to two decimal places.

Find the indefinite integral. $\int \frac { 3 x ^ { 2 } } { e ^ { x } } d x$

Use Simpson's Rule to approximate the revenue for the marginal revenue function $\frac { d R } { d x } = 5 \sqrt { 8000 - x ^ { 3 } }$ with n = 4. Assume that the number of units sold, x, increases from 14 to 18. Round your answer to one decimal place.

Evaluate the improper integral if it converges, or state that it diverges. $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 8 ] { x } } d x$

Use a table of integrals to find the indefinite integral $\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x$ .

Find the indefinite integral. $\int \frac { 6 ( \ln x ) ^ { 2 } } { x ^ { 2 } } d x$

Decide whether the following integral is improper. $\int _ { 0 } ^ { 1 } \frac { 1 } { 3 x - 2 } d x$

Identify u and dv for finding the integral using integration by parts. $\int x ^ { 3 } \ln 9 x$ dx

Find the indefinite integral. $\int \frac { t } { \sqrt { 1 + 7 t } } d t$

Use a table of integrals with forms involving $e ^ { u }$ to find the integral. $\int \frac { - 7 } { 1 + e ^ { - 6 x } } d x$

A body assimilates a 12-hour cold tablet at a rate modeled by $d C / d t = 8 - \ln \left( t ^ { 2 } - 2 t + 4 \right) , 0 \leq t \leq 12$ where $d C / d t$ is measured in milligrams per hour and $t$ is the time in hours. Use Simpson's Rule with $n = 8$ to estimate the total amount of the drug absorbed into the body during the 12 hours.

Find the indefinite integral. $\int \frac { \ln v } { 9 v ^ { 3 } } d v$

The revenue (in dollars per year) for a new product is modeled by $R = 10,000 \left[ 1 - \frac { 1 } { \left( 1 + 0.1 ^ { 2 } \right) ^ { 1 / 2 } } \right]$ where t the time in years. Estimate the total revenue from sales of the product over its first 4 years on the market. Round your answer to nearest dollar

Use a table of integrals with forms involving $\sqrt { a ^ { 2 } - u ^ { 2 } }$ to find $\int \frac { - 3 } { x ^ { 2 } \sqrt { 49 - x ^ { 2 } } } d x$

Use a table of integrals to find the indefinite integral $\int ( \ln 3 x ) ^ { 2 } d x$ .

Evaluate the definite integral $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { 2 x } d x$ . Round your answer to three decimal places.

Find the integral below using an integral table. $\int \frac { 6 } { 64 - x ^ { 2 } } d x$

Approximate the definite integral "by hand," using the Trapezoidal Rule with $n = 4$ trapezoids. Round answer to three decimal places. $\int _1^ { 2 } \frac { 7 } { x } d x$

Find the indefinite integral. $\int v \ln ( v + 2 ) d v$

Approximate the integral using Simpson's Rule: $\int _ { 0 } ^ { 5 } \frac { x } { 2 + x + x ^ { 2 } } d x$ , n = 6. Round your answer to three decimal places.