Exam 5: Antiderivatives and Indefinite Integration

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. $\int _ { 0 } ^ { 5 } ( 6 t + 1 ) d t$

Find the indefinite integral. $\int \frac { x ^ { 2 } + 16 x + 6 } { x ^ { 3 } + 24 x ^ { 2 } + 18 x - 1 } d x$

Determine the area of the given region. $y = 4 x ( 1 - x )$

Write the limit $\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } 4 c _ { i } \left( 7 - c _ { i } \right) ^ { 2 } \Delta x _ { i } , \text { as a definite integral on the interval }$ $[ 0,8 ]$ where $c _ { i }$ is any point in the $i ^ { t h }$ subinterval.

Find the indefinite integral $\int 10 \sin s + 7 \cos s d s$

Evaluate the integral. $\int _ { 3 } ^ { 4 } \left( 24 x ^ { 2 } - 1 \right) d x$ given, $\int _ { 3 } ^ { 4 } x ^ { 3 } d x = \frac { 175 } { 4 }$ $\int _ { 3 } ^ { 4 } x ^ { 2 } d x = \frac { 37 } { 3 }$ $\int _ { 3 } ^ { 4 } x d x = \frac { 7 } { 2 }$ 4 $\int _ { 3 } ^ { 4 } d x = 1$

Find the indefinite integral. $\int 2 x e ^ { 4 x ^ { 2 } } d x$

A ball is thrown vertically upwards from a height of ft with an initial velocity of $30 \mathrm { ft }$ per second. How high will the ball go? Note that the acceleration of the ball is given by $a ( t ) = - 32$ feet per second per second.

Evaluate $\int _ { 1 } ^ { e } \frac { ( 1 + \ln x ) ^ { 5 } } { x } d x$

Find the indefinite integral $\int ( - 8 t + 7 ) d t$

Find the area of the region bounded by the graphs of the equations $y = \frac { x + 2 } { x } , x = 1 , x = 5$ , and $y = 0$ . Round your answer to four decimal places.

Use Simpson's Rule to approximate the value of the definite integral $\int_{0}^{\frac{\pi}{4}} 6 x \tan 2 x d x$ with $n = 4$ . Round your answer to four decimal places.

$\text { Find the integral } \int \frac { x - 15 } { x ^ { 2 } + 1 } d x \text {. }$

$\text { Find } \int \tan 6 \theta d \theta \text {. }$

Evaluate the definite integral of the algebraic function. $\int _ { 3 } ^ { 6 } ( 5 u + 4 ) d u$ Use a graphing utility to verify your results.

Evaluate the definite integral. $\int _ { n 2 } ^ { \operatorname { m } 4 } e ^ { - x } d x$

$\text { Find } F ^ { \prime } ( x ) \text { given }$ $F ( x ) = \int _ { x } ^ { x + 3 } ( 10 t + 1 ) d t$

Find the indefinite integral. $\int \frac { 1 } { 36 + ( x - 4 ) ^ { 2 } } d x$

Use the summation formulas to rewrite the expression $\sum _ { k = 1 } ^ { n } \frac { 24 k ( k - 1 ) } { n ^ { 7 } }$ without the summation notation.

Evaluate the integral. $\int _ { 7 } ^ { 8 } ( - 24 s + 1 ) d s$ given, $\int ^ { 8 } x ^ { 3 } d x = \frac { 1695 } { 4 } \text {, }$ $\int x ^ { 2 } d x = \frac { 169 } { 3 }$ $\int x d x = \frac { 15 } { 2 } ,$ $\int d x = 1$