Exam 5: Antiderivatives and Indefinite Integration

The height above the ground of an object thrown upward from a point feet above the ground with an initial velocity of feet per second is given by the function $f ( t ) = - 16 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ . A balloon, rising vertically with a velocity of 24 feet per second, releases a sandbag at the instant it is 60 feet above the ground. How many seconds after its release will the bag strike the ground? Round your answer to three decimal places.

$\text { Find } \int \frac { \sqrt { x } } { x - 4 } d x \text {. }$

$\text { Find the limit of } s ( n ) \text { as } n \rightarrow \infty \text {. }$ $s ( n ) = \frac { 1 } { n ^ { 2 } } \left[ \frac { n ( n + 1 ) ( 5 n + 1 ) } { 10 } \right]$

$\text { Find } F ^ { t } ( x ) \text { given }$ $F ( x ) = \int _ { - 2 x } ^ { 2 x } s ^ { 2 } d s$

$\text { Find the area of the region given below bounded by the graph of } y = \frac { 6 } { x } \text {. }$

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 1 } 2 \sin \left( x ^ { 2 } \right) d x$ is less than $0.00001$ using the Trapezoidal Rule. Use a graphing utility to estimate the maximum of the absolute value of the second derivative.

$\text { Find the integral } \int \frac { t } { t ^ { 4 } + 81 } d t \text {. }$

$\text { Use Simpson's Rule with } n = 14 \text { to approximate } \pi \text { using the equation }$ $\pi = \int _ { 0 } ^ { 1 } \frac { 4 } { 1 + x ^ { 2 } } d x$ . Round your answer to five decimal places.

$\text { Find the integral } \int \frac { \sin x } { 13 + \cos ^ { 2 } x } d x \text {. }$

Find the indefinite integral. $\int \ln \left( \mathrm { e } ^ { 16 x ^ { 15 } - 2 } \right) d x$

$\text { Find the solution of the differential equation } \frac { d r } { d t } = \frac { \sec ^ { 2 } t } { \tan t + 1 } \text { which passes through the }$ point $( \pi , 5 )$

Find the limit of $s ( n ) \text { as } n \rightarrow \infty$ $s ( n ) = \frac { 5 } { n ^ { 3 } } \left( \frac { n ^ { 3 } ( n + 1 ) ^ { 2 } } { 7 } \right]$

Find the indefinite integral. $\int \csc ( 4 x ) d x$

$\text { Use the differential equation } \frac { d y } { d x } = \frac { 1 } { 81 + x ^ { 2 } } \text { and the initial condition } y ( 9 ) = \pi \text { to }$ find y.

The oscillating current in an electrical circuit is $I = 2 \sin ( 60 \pi t ) + \cos ( 120 \pi t ) \text { where } I$ is measured in amperes and t is measured in seconds. Find the average current for the time interval $0 \leq t \leq \frac { 1 } { 120 }$ . Round your answer to three decimal places.

Determine the area of the given region. $y = x + \sin x , 0 \leq x \leq \pi$

$\text { Suppose a population of bacteria is changing at a rate of } \frac { d p } { d t } = \frac { 4,000 } { 1 + 0.5 t } \text {, where } t \text { is }$ the time in days. The initial population (when $t = 0$ ) is 1,500 . Identify an equation that gives the population at any time $t$ and use it to find the population when $t = 5$ days. Round your answer to the nearest integer.

Find the indefinite integral $\int \left( 12 s ^ { 3 } - 12 s + 3 \right) d s$

Use the limit process to find the area of the region between the graph of the function $y = 16 - x ^ { 2 }$ and $x$ -axis over the interval $[ - 4,4 ]$

$\text { Find the area of the shaded region for the function } y = \frac { 4 } { x ^ { 2 } + 4 x + 8 } \text {. }$