Question 1

Find the indefinite integral. $$\int \frac { 1 } { x \sqrt { 64 x ^ { 2 } - 49 } } d x$$

Accepted Answer

A)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 8 | x } { 7 } \right) + C$ 
B) 
$7 \operatorname { arcsec } \left( \frac { | 7 | x } { 8 } \right) + C$ 
C)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 64 | x } { 49 } \right) + C$ 
D)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 7 | x } { 8 } \right) + C$
C.
$7 \operatorname { arcsec } \left( \frac { | 64 | x } { 49 } \right) + C$ 
A)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 8 | x } { 7 } \right) + C$ 
B) 
$7 \operatorname { arcsec } \left( \frac { | 7 | x } { 8 } \right) + C$ 
C)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 64 | x } { 49 } \right) + C$ 
D)  $\frac { 1 } { 7 } \operatorname { arcsec } \left( \frac { | 7 | x } { 8 } \right) + C$
C.
$7 \operatorname { arcsec } \left( \frac { | 64 | x } { 49 } \right) + C$

Question 2

Find  the indefinite integral of the following function. 
$\int \pi \sin 5 z d z$

Accepted Answer

A)  $\frac { - \cos 5 \pi } { 6 } + C$ 
B)  $\frac { - \cos 5 \pi z } { 5 } + C$ 
C)  $\frac { \sin 5 \pi z } { 5 } + C$ 
D)  $\frac { - \cos 5 z } { 5 } + C$ 
E)  $- 5 \cos 5 \pi + C$ 
A)  $\frac { - \cos 5 \pi } { 6 } + C$ 
B)  $\frac { - \cos 5 \pi z } { 5 } + C$ 
C)  $\frac { \sin 5 \pi z } { 5 } + C$ 
D)  $\frac { - \cos 5 z } { 5 } + C$ 
E)  $- 5 \cos 5 \pi + C$

Question 3

$\text { Suppose the demand equation for a product is } p = \frac { 50,000 } { 500 + 7 x } \text {. Find the average price }$ $p$ on the interval $50 \leq x \leq 60$. Round your answer to the nearest dollar.

Accepted Answer

A)  $\$ 67$ 
B)  $\$ 83$ 
C)  $\$ 57$ 
D)  $\$ 565$ 
E)  $\$ 52$ 
A)  $\$ 67$ 
B)  $\$ 83$ 
C)  $\$ 57$ 
D)  $\$ 565$ 
E)  $\$ 52$

Question 4

Evaluate the definite integral of the algebraic function. $\int _ { 5 } ^ { 9 } 9 \left| z ^ { 2 } - 36 \right| d z$ Use a graphing utility to verify your results.

Accepted Answer

A)  $- 516$ 
B)  516 
C)  618 
D)  1185 
E)  $- 1974$ 
A)  $- 516$ 
B)  516 
C)  618 
D)  1185 
E)  $- 1974$

Question 5

Write the following limit as a definite integral on the interval [2 , 5], where ci is any point in the ith subinterval. $\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } \left( - 7 c _ { i } + 2 \right) \Delta x _ { i }$

Accepted Answer

A)  $\int _ { 2 } ^ { 5 } \left( - 7 x ^ { 2 } + 2 x \right) d x$ 
B)  $\int _ { 2 } ^ { 5 } \left( - \frac { 7 } { 2 } x ^ { 2 } - 2 x \right) d x$ 
C)  $\int _ { 2 } ^ { 5 } ( - 7 x + 2 ) d x$ 
D) $\int _ { 5 } ^ { 2 } ( - 7 x + 2 ) d x$ 
E)  $\int _ { 2 } ^ { 5 } ( - 7 x - 2 ) d x$ 
A)  $\int _ { 2 } ^ { 5 } \left( - 7 x ^ { 2 } + 2 x \right) d x$ 
B)  $\int _ { 2 } ^ { 5 } \left( - \frac { 7 } { 2 } x ^ { 2 } - 2 x \right) d x$ 
C)  $\int _ { 2 } ^ { 5 } ( - 7 x + 2 ) d x$ 
D) $\int _ { 5 } ^ { 2 } ( - 7 x + 2 ) d x$ 
E)  $\int _ { 2 } ^ { 5 } ( - 7 x - 2 ) d x$

Question 6

Find the indefinite integral $\int \left( 12 u ^ { 2 } + 4 u \right) d u$

Accepted Answer

A)  $4 u ^ { 3 } + 2 u ^ { 2 } + C$ 
B)  $12 u ^ { 3 } + 4 u ^ { 2 } + C$ 
C)  $12 u ^ { 3 } + 2 u ^ { 2 } + C$ 
D)  $4 u ^ { 3 } + 4 u ^ { 2 } + C$ 
E)  $24 u + 4 + C$ 
A)  $4 u ^ { 3 } + 2 u ^ { 2 } + C$ 
B)  $12 u ^ { 3 } + 4 u ^ { 2 } + C$ 
C)  $12 u ^ { 3 } + 2 u ^ { 2 } + C$ 
D)  $4 u ^ { 3 } + 4 u ^ { 2 } + C$ 
E)  $24 u + 4 + C$

Question 7

$\text { Use the Trapezoid Rule to approximate the value of the definite integral } \int _ { 0 } ^ { 2 } x ^ { 4 } d x \text { wth }$ $n = 4$. 
Round your answer to four decimal places.

Accepted Answer

A)  $7.0625$ 
B)  $5.7813$ 
C)  $7.0313$ 
D)  $6.5625$ 
E)  $28.2500$ 
A)  $7.0625$ 
B)  $5.7813$ 
C)  $7.0313$ 
D)  $6.5625$ 
E)  $28.2500$

Question 8

Find the indefinite integral. $\int \frac { \cos ( 2 \theta ) } { \sin ( 2 \theta ) } d \theta$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln | \tan ( 2 \theta ) | + C$ 
B)  $\frac { 1 } { 2 } \ln | \sin ( 2 \theta ) | + C$ 
C)  $\frac { 1 } { 2 } \ln | \cos ( 2 \theta ) | + C$ 
D)  $2 \ln | \sin ( 2 \theta ) | + C$ 
E)  $2 \ln | \cos ( 2 \theta ) | + C$ 
A)  $\frac { 1 } { 2 } \ln | \tan ( 2 \theta ) | + C$ 
B)  $\frac { 1 } { 2 } \ln | \sin ( 2 \theta ) | + C$ 
C)  $\frac { 1 } { 2 } \ln | \cos ( 2 \theta ) | + C$ 
D)  $2 \ln | \sin ( 2 \theta ) | + C$ 
E)  $2 \ln | \cos ( 2 \theta ) | + C$

Question 9

Use the properties of summation and Theorem 5.2 to evaluate the sum. $$\sum _ { i = 1 } ^ { 37 } ( i - 6 ) ^ { 2 }$$

Accepted Answer

A)  $9,176$ 
B)  10,471 
C)  18,907 
D)  $9,139$ 
E)  98,346 
A)  $9,176$ 
B)  10,471 
C)  18,907 
D)  $9,139$ 
E)  98,346

Question 10

Find the indefinite integral of the following function. $\int \cos 4 z d z$

Accepted Answer

A)  $\cos 4 z + C$ 
B)  $\frac { \sin 4 z } { 4 } + C$ 
C)  $4 \sin 4 z + C$ 
D)  $\sin 4 z + C$ 
E)  $\frac { \sin 4 z } { 5 } + C$ 
A)  $\cos 4 z + C$ 
B)  $\frac { \sin 4 z } { 4 } + C$ 
C)  $4 \sin 4 z + C$ 
D)  $\sin 4 z + C$ 
E)  $\frac { \sin 4 z } { 5 } + C$

Question 11

Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 8$ when approximating the following integral.
$$\int _ { 4 } ^ { 6 } ( 6 z + 3 ) d z$$

Accepted Answer

A)  The error for the Trapezoidal Rule is $0.0051$ and for Simpson's Rule it is $0.0013$. 
B)  The error for the Trapezoidal Rule is $0.0000$ and for Simpson's Rule it is $0.0200$. 
C)  The error for the Trapezoidal Rule is $0.0204$ and for Simpson's Rule it is $0.0200$. 
D)  The error for the Trapezoidal Rule is $0.0204$ and for Simpson's Rule it is $0.0000$. 
E)  The error for the Trapezoidal Rule is $0.0000$ and for Simpson's Rule it is $0.0000$. 
A)  The error for the Trapezoidal Rule is $0.0051$ and for Simpson's Rule it is $0.0013$. 
B)  The error for the Trapezoidal Rule is $0.0000$ and for Simpson's Rule it is $0.0200$. 
C)  The error for the Trapezoidal Rule is $0.0204$ and for Simpson's Rule it is $0.0200$. 
D)  The error for the Trapezoidal Rule is $0.0204$ and for Simpson's Rule it is $0.0000$. 
E)  The error for the Trapezoidal Rule is $0.0000$ and for Simpson's Rule it is $0.0000$.

Question 12

Use the properties of summation and Theorem 5.2 to evaluate the sum. $$\sum _ { i = 1 } ^ { 22 } i \left( i ^ { 2 } + 8 \right)$$

Accepted Answer

A)  2,156 
B)  64,262 
C)  68,057 
D)  258,060 
E)  66,033 
A)  2,156 
B)  64,262 
C)  68,057 
D)  258,060 
E)  66,033

Question 13

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 5 } \sqrt { 6 + x } d x$ is less than $0.00001$ using Simpson's Rule.

Accepted Answer

A)  10 
B)  16 
C)  6 
D)  8 
E)  13 
A)  10 
B)  16 
C)  6 
D)  8 
E)  13

Question 14

$\text { Find the integral } \int \frac { 15 } { 2 \sqrt { x } ( 1 + x ) } d x \text {. }$

Accepted Answer

A)  $\frac { 30 } { \arctan ( \sqrt { x } ) } + C$ 
B)  $15 \arctan ( \sqrt { x } ) + C$ 
C)  $\frac { 15 } { 2 \arctan ( \sqrt { x } ) } + C$ 
D)  $\quad 30 \arctan ( \sqrt { x } ) + C$ 
E)  $\frac { 15 } { \arctan ( \sqrt { x } ) } + C$ 
A)  $\frac { 30 } { \arctan ( \sqrt { x } ) } + C$ 
B)  $15 \arctan ( \sqrt { x } ) + C$ 
C)  $\frac { 15 } { 2 \arctan ( \sqrt { x } ) } + C$ 
D)  $\quad 30 \arctan ( \sqrt { x } ) + C$ 
E)  $\frac { 15 } { \arctan ( \sqrt { x } ) } + C$

Question 15

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 20 feet above the ground. At what velocity will it hit the ground? Round your answer to three decimal places.

Accepted Answer

A)  $- 60.111 \mathrm { ft } / \mathrm { sec }$ 
B)  $- 43.081 \mathrm { ft } / \mathrm { sec }$ 
C)  $- 15.028 \mathrm { ft } / \mathrm { sec }$ 
D)  $49.298 \mathrm { ft } / \mathrm { sec }$ 
E)  $67.081 \mathrm { ft } / \mathrm { sec }$ 
A)  $- 60.111 \mathrm { ft } / \mathrm { sec }$ 
B)  $- 43.081 \mathrm { ft } / \mathrm { sec }$ 
C)  $- 15.028 \mathrm { ft } / \mathrm { sec }$ 
D)  $49.298 \mathrm { ft } / \mathrm { sec }$ 
E)  $67.081 \mathrm { ft } / \mathrm { sec }$

Question 16

$\text { Suppose a population of bacteria is changing at a rate of } \frac { d p } { d t } = \frac { 1,000 } { 1 + 0.5 t } \text {, where } t \text { is }$ the time in days. The initial population (when $t = 0$ ) is 500 . Write an equation that gives the population at any time $t$.

Accepted Answer

A)  $p ( t ) = 2,000 \ln | 1 + 0.5 t | + 250$ 
B)  $p ( t ) = \ln | 1 + 0.5 t | + 500$ 
C)  $p ( t ) = 1,000 \ln | 1 + 0.5 t | + 500$ 
D)  $p ( t ) = 1,000 \ln | 1 + 0.5 t | + 250$ 
E)  $p ( t ) = 2,000 \ln | 1 + 0.5 t | + 500$ 
A)  $p ( t ) = 2,000 \ln | 1 + 0.5 t | + 250$ 
B)  $p ( t ) = \ln | 1 + 0.5 t | + 500$ 
C)  $p ( t ) = 1,000 \ln | 1 + 0.5 t | + 500$ 
D)  $p ( t ) = 1,000 \ln | 1 + 0.5 t | + 250$ 
E)  $p ( t ) = 2,000 \ln | 1 + 0.5 t | + 500$

Question 17

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 4 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 4 } ^ { 7 } \frac { 1 } { \sqrt { t } \sqrt { 9 - t } } d t$

Accepted Answer

A)  The Trapezoidal rule gives $0.7100$ and Simpson's rule gives $0.7005$. 
B)  The Trapezoidal rule gives $- 0.7028$ and Simpson's rule gives $- 0.7005$. 
C)  The Trapezoidal rule gives $0.7100$ and Simpson's rule gives $0.7380$. 
D)  The Trapezoidal rule gives $0.7028$ and Simpson's rule gives $0.7005$. 
E)  The Trapezoidal rule gives $0.7028$ and Simpson's rule gives $0.7380$. 
A)  The Trapezoidal rule gives $0.7100$ and Simpson's rule gives $0.7005$. 
B)  The Trapezoidal rule gives $- 0.7028$ and Simpson's rule gives $- 0.7005$. 
C)  The Trapezoidal rule gives $0.7100$ and Simpson's rule gives $0.7380$. 
D)  The Trapezoidal rule gives $0.7028$ and Simpson's rule gives $0.7005$. 
E)  The Trapezoidal rule gives $0.7028$ and Simpson's rule gives $0.7380$.

Question 18

Find the indefinite integral of the following function and check the result by differentiation. $$\int \frac { 5 s } { \left( s ^ { 2 } + 3 \right) ^ { 4 } } d s$$

Accepted Answer

A)  $\frac { 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 3 } } + C$ 
B)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 3 } } + C$ 
C)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 5 } } + C$ 
D)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { - 3 } } + C$ 
E)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 4 } } + C$ 
A)  $\frac { 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 3 } } + C$ 
B)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 3 } } + C$ 
C)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 5 } } + C$ 
D)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { - 3 } } + C$ 
E)  $\frac { - 5 } { 6 \left( s ^ { 2 } + 3 \right) ^ { 4 } } + C$

Question 19

Solve the differential equation. $$\frac { d f } { d s } = 12 s ^ { 3 } , \quad f ( 1 ) = - 2$$

Accepted Answer

A)  $f ( s ) = 3 s ^ { 4 } + 5$ 
B)  $f ( s ) = 3 s ^ { 4 } + 4 s$ 
C)  $f ( s ) = 3 s ^ { 4 } - 5$ 
D)  $f ( s ) = 3 s ^ { 4 } + C$ 
E)  $f ( s ) = 36 s ^ { 2 } + 5$ 
A)  $f ( s ) = 3 s ^ { 4 } + 5$ 
B)  $f ( s ) = 3 s ^ { 4 } + 4 s$ 
C)  $f ( s ) = 3 s ^ { 4 } - 5$ 
D)  $f ( s ) = 3 s ^ { 4 } + C$ 
E)  $f ( s ) = 36 s ^ { 2 } + 5$

Question 20

Evaluate the following definite integral. $\int _ { 0 } ^ { 5 \sqrt { \pi } } z \cos \left( 7 z ^ { 2 } \right) d z$Use a graphing utility to check your answer.

Accepted Answer

A)  $- \frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  $- 1$ 
D)  1 
E)  0 
A)  $- \frac { 1 } { 14 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  $- 1$ 
D)  1 
E)  0

Find the indefinite integral. $\int \frac { 1 } { x \sqrt { 64 x ^ { 2 } - 49 } } d x$

Find the indefinite integral of the following function. $\int \pi \sin 5 z d z$

$\text { Suppose the demand equation for a product is } p = \frac { 50,000 } { 500 + 7 x } \text {. Find the average price }$ $p$ on the interval $50 \leq x \leq 60$ . Round your answer to the nearest dollar.

Evaluate the definite integral of the algebraic function. $\int _ { 5 } ^ { 9 } 9 \left| z ^ { 2 } - 36 \right| d z$ Use a graphing utility to verify your results.

Write the following limit as a definite integral on the interval [2 , 5], where ci is any point in the ith subinterval. $\lim _ { | \Delta x | \rightarrow 0 } \sum _ { i = 1 } ^ { n } \left( - 7 c _ { i } + 2 \right) \Delta x _ { i }$

Find the indefinite integral $\int \left( 12 u ^ { 2 } + 4 u \right) d u$

$\text { Use the Trapezoid Rule to approximate the value of the definite integral } \int _ { 0 } ^ { 2 } x ^ { 4 } d x \text { wth }$ $n = 4$ . Round your answer to four decimal places.

Find the indefinite integral. $\int \frac { \cos ( 2 \theta ) } { \sin ( 2 \theta ) } d \theta$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 37 } ( i - 6 ) ^ { 2 }$

Find the indefinite integral of the following function. $\int \cos 4 z d z$

Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 8$ when approximating the following integral. $\int _ { 4 } ^ { 6 } ( 6 z + 3 ) d z$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 22 } i \left( i ^ { 2 } + 8 \right)$

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 5 } \sqrt { 6 + x } d x$ is less than $0.00001$ using Simpson's Rule.

$\text { Find the integral } \int \frac { 15 } { 2 \sqrt { x } ( 1 + x ) } d x \text {. }$

$\text { Suppose a population of bacteria is changing at a rate of } \frac { d p } { d t } = \frac { 1,000 } { 1 + 0.5 t } \text {, where } t \text { is }$ the time in days. The initial population (when $t = 0$ ) is 500 . Write an equation that gives the population at any time $t$ .

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 4 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. $\int _ { 4 } ^ { 7 } \frac { 1 } { \sqrt { t } \sqrt { 9 - t } } d t$

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 5 s } { \left( s ^ { 2 } + 3 \right) ^ { 4 } } d s$

Solve the differential equation. $\frac { d f } { d s } = 12 s ^ { 3 } , \quad f ( 1 ) = - 2$

Evaluate the following definite integral. $\int _ { 0 } ^ { 5 \sqrt { \pi } } z \cos \left( 7 z ^ { 2 } \right) d z$ Use a graphing utility to check your answer.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 5: Antiderivatives and Indefinite Integration

Find the indefinite integral. ∫1x64x2−49dx\int \frac { 1 } { x \sqrt { 64 x ^ { 2 } - 49 } } d x∫x64x2−49​1​dx

Find the indefinite integral of the following function. ∫πsin⁡5zdz\int \pi \sin 5 z d z∫πsin5zdz

Evaluate the definite integral of the algebraic function. ∫599∣z2−36∣dz\int _ { 5 } ^ { 9 } 9 \left| z ^ { 2 } - 36 \right| d z∫59​9​z2−36​dz Use a graphing utility to verify your results.

Find the indefinite integral ∫(12u2+4u)du\int \left( 12 u ^ { 2 } + 4 u \right) d u∫(12u2+4u)du

Find the indefinite integral. ∫cos⁡(2θ)sin⁡(2θ)dθ\int \frac { \cos ( 2 \theta ) } { \sin ( 2 \theta ) } d \theta∫sin(2θ)cos(2θ)​dθ

Use the properties of summation and Theorem 5.2 to evaluate the sum. ∑i=137(i−6)2\sum _ { i = 1 } ^ { 37 } ( i - 6 ) ^ { 2 }∑i=137​(i−6)2

Find the indefinite integral of the following function. ∫cos⁡4zdz\int \cos 4 z d z∫cos4zdz

Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule with n=8n = 8n=8 when approximating the following integral. ∫46(6z+3)dz\int _ { 4 } ^ { 6 } ( 6 z + 3 ) d z∫46​(6z+3)dz

Use the properties of summation and Theorem 5.2 to evaluate the sum. ∑i=122i(i2+8)\sum _ { i = 1 } ^ { 22 } i \left( i ^ { 2 } + 8 \right)∑i=122​i(i2+8)

Find the smallest n such that the error estimate in the approximation of the definite integral ∫056+xdx\int _ { 0 } ^ { 5 } \sqrt { 6 + x } d x∫05​6+x​dx is less than 0.000010.000010.00001 using Simpson's Rule.

Find the integral ∫152x(1+x)dx. \text { Find the integral } \int \frac { 15 } { 2 \sqrt { x } ( 1 + x ) } d x \text {. } Find the integral ∫2x​(1+x)15​dx.

Find the indefinite integral of the following function and check the result by differentiation. ∫5s(s2+3)4ds\int \frac { 5 s } { \left( s ^ { 2 } + 3 \right) ^ { 4 } } d s∫(s2+3)45s​ds

Solve the differential equation. dfds=12s3,f(1)=−2\frac { d f } { d s } = 12 s ^ { 3 } , \quad f ( 1 ) = - 2dsdf​=12s3,f(1)=−2

Evaluate the following definite integral. ∫05πzcos⁡(7z2)dz\int _ { 0 } ^ { 5 \sqrt { \pi } } z \cos \left( 7 z ^ { 2 } \right) d z∫05π​​zcos(7z2)dz Use a graphing utility to check your answer.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Find the indefinite integral. $\int \frac { 1 } { x \sqrt { 64 x ^ { 2 } - 49 } } d x$

Find the indefinite integral of the following function. $\int \pi \sin 5 z d z$

Evaluate the definite integral of the algebraic function. $\int _ { 5 } ^ { 9 } 9 \left| z ^ { 2 } - 36 \right| d z$ Use a graphing utility to verify your results.

Find the indefinite integral $\int \left( 12 u ^ { 2 } + 4 u \right) d u$

Find the indefinite integral. $\int \frac { \cos ( 2 \theta ) } { \sin ( 2 \theta ) } d \theta$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 37 } ( i - 6 ) ^ { 2 }$

Find the indefinite integral of the following function. $\int \cos 4 z d z$

Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule with $n = 8$ when approximating the following integral. $\int _ { 4 } ^ { 6 } ( 6 z + 3 ) d z$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 22 } i \left( i ^ { 2 } + 8 \right)$

Find the smallest n such that the error estimate in the approximation of the definite integral $\int _ { 0 } ^ { 5 } \sqrt { 6 + x } d x$ is less than $0.00001$ using Simpson's Rule.

$\text { Find the integral } \int \frac { 15 } { 2 \sqrt { x } ( 1 + x ) } d x \text {. }$

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 5 s } { \left( s ^ { 2 } + 3 \right) ^ { 4 } } d s$

Solve the differential equation. $\frac { d f } { d s } = 12 s ^ { 3 } , \quad f ( 1 ) = - 2$

Evaluate the following definite integral. $\int _ { 0 } ^ { 5 \sqrt { \pi } } z \cos \left( 7 z ^ { 2 } \right) d z$ Use a graphing utility to check your answer.