Question 1

Evaluate the following definite integral. $\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 6 t + 3 } } d t$ Use a graphing utility to check your answer.

Accepted Answer

A)  $\frac { \sqrt { 33 } - \sqrt { 21 } } { 6 }$ 
B)  $\frac { \sqrt { 21 } - \sqrt { 33 } } { 3 }$ 
C)  $\frac { \sqrt { 33 } + \sqrt { 21 } } { 6 }$ 
D)  $\frac { \sqrt { 33 } + \sqrt { 21 } } { 3 }$ 
E)  $\frac { \sqrt { 33 } - \sqrt { 21 } } { 3 }$ 
A)  $\frac { \sqrt { 33 } - \sqrt { 21 } } { 6 }$ 
B)  $\frac { \sqrt { 21 } - \sqrt { 33 } } { 3 }$ 
C)  $\frac { \sqrt { 33 } + \sqrt { 21 } } { 6 }$ 
D)  $\frac { \sqrt { 33 } + \sqrt { 21 } } { 3 }$ 
E)  $\frac { \sqrt { 33 } - \sqrt { 21 } } { 3 }$

Question 2

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x$

Accepted Answer

A)  The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.435222$. 
B)  The Trapezoidal rule gives $- 1.366878$ and Simpson's rule gives $- 1.366906$. 
C)  The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.366906$. 
D)  The Trapezoidal rule gives $1.366795$ and Simpson's rule gives $1.435222$. 
E)  The Trapezoidal rule gives $1.366795$ and Simpson's rule gives 1.366906. 
A)  The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.435222$. 
B)  The Trapezoidal rule gives $- 1.366878$ and Simpson's rule gives $- 1.366906$. 
C)  The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.366906$. 
D)  The Trapezoidal rule gives $1.366795$ and Simpson's rule gives $1.435222$. 
E)  The Trapezoidal rule gives $1.366795$ and Simpson's rule gives 1.366906.

Question 3

Use the Trapezoidal Rule to approximate the value of the definite integral $ \int_{\frac{\pi}{2}}^{x} 5 \sqrt{x} \sin 4 x d x $
with $n = 4$. Round your answer to four decimal places. $\frac { x } { 2 }$

Accepted Answer

A)  $- 8.0654$ 
B)  $- 4.0327$ 
C)  $- 2.0164$ 
D)  $- 1.0082$ 
E)  $- 0.5041$ 
A)  $- 8.0654$ 
B)  $- 4.0327$ 
C)  $- 2.0164$ 
D)  $- 1.0082$ 
E)  $- 0.5041$

Question 4

Find the area of the shaded region. $y = \frac { 1 } { 4 } e ^ { x }$

Accepted Answer

A) $\frac { 1 } { 4 }$ 
B)  $4 e$ 
C)  1 
D)  $\frac { e } { 4 }$ 
E)  $\frac { e - 1 } { 4 }$ 
A) $\frac { 1 } { 4 }$ 
B)  $4 e$ 
C)  1 
D)  $\frac { e } { 4 }$ 
E)  $\frac { e - 1 } { 4 }$

Question 5

$\text { Use the differential equation } \frac { d y } { d x } = \frac { 6 } { \sqrt { 64 - x ^ { 2 } } } \text { and the initial condition } y ( 0 ) = \pi \text { to }$ find y.

Accepted Answer

A)  $y = 6 \arccos ( 64 x ) + \pi$ 
B)  $y = 6 \arcsin \left( \frac { 1 } { 8 } x \right) + \pi$ 
C)  $y = 6 \arccos \left( \frac { 1 } { 8 } x \right) + \pi$ 
D) $y = 6 \arccos \left( \frac { 1 } { 64 } x \right) + \pi$ 
E)  $y = 6 \arcsin ( 8 x ) + \pi$ 
A)  $y = 6 \arccos ( 64 x ) + \pi$ 
B)  $y = 6 \arcsin \left( \frac { 1 } { 8 } x \right) + \pi$ 
C)  $y = 6 \arccos \left( \frac { 1 } { 8 } x \right) + \pi$ 
D) $y = 6 \arccos \left( \frac { 1 } { 64 } x \right) + \pi$ 
E)  $y = 6 \arcsin ( 8 x ) + \pi$

Question 6

$\text { Use integration to find the particular solution of the differential equation } \frac { d y } { d x } = \frac { \ln x } { x }$ which passes through the point $( 1 , - 3 )$.

Accepted Answer

A)  $y = \ln x + 3$ 
B) $y = \frac { ( \ln x ) ^ { 2 } } { 2 } - 3$ 
C) $y = \frac { ( \ln x ) ^ { 2 } } { 2 } + 3$ 
D)  $y = - 3 ( \ln x + 1 )$ 
E)  $y = \ln x - 3$\ 
A)  $y = \ln x + 3$ 
B) $y = \frac { ( \ln x ) ^ { 2 } } { 2 } - 3$ 
C) $y = \frac { ( \ln x ) ^ { 2 } } { 2 } + 3$ 
D)  $y = - 3 ( \ln x + 1 )$ 
E)  $y = \ln x - 3$\

Question 7

Evaluate the definite integral of the algebraic function. $$\int _ { 0 } ^ { 5 } 4 | x - 4 | d x$$ Use a graphing utility to verify your results.

Accepted Answer

A)  100 
B)  $- 30$ 
C)  196 
D)  $- 99$ 
E)  98 
A)  100 
B)  $- 30$ 
C)  196 
D)  $- 99$ 
E)  98

Question 8

$\text { Find } F ^ {t } ( x ) \text { if } F ( x ) = \int _ { 1 } ^ { 14 x ^ { 2 } } \frac { 1 } { t } d t$

Accepted Answer

A)  $F ^ { t } ( x ) = \frac { 1 } { 14 x ^ { 2 } }$ 
B)  $F ^ { t } ( x ) = \frac { 1 } { 14 x ^ { 2 } - 1 }$ 
C)  $F ^ {t } ( x ) = \frac { 14 } { x }$ 
D)  $F ^ { t } ( x ) = \frac { 28 } { x }$ 
E)  $F ^ { t} ( x ) = \frac { 2 } { x }$ 
A)  $F ^ { t } ( x ) = \frac { 1 } { 14 x ^ { 2 } }$ 
B)  $F ^ { t } ( x ) = \frac { 1 } { 14 x ^ { 2 } - 1 }$ 
C)  $F ^ {t } ( x ) = \frac { 14 } { x }$ 
D)  $F ^ { t } ( x ) = \frac { 28 } { x }$ 
E)  $F ^ { t} ( x ) = \frac { 2 } { x }$

Question 9

Use the properties of summation and Theorem 5.2 to evaluate the sum. $$\sum _ { i = 1 } ^ { 34 } ( 4 i - 8 )$$

Accepted Answer

A)  1156 
B)  1190 
C)  2108 
D)  2244 
E)  323 
A)  1156 
B)  1190 
C)  2108 
D)  2244 
E)  323

Question 10

Solve the differential equation. $\frac { d P } { d x } = 15 x ^ { 4 } - 5 , \quad P ( - 2 ) = 8$

Accepted Answer

A)  $P ( x ) = 3 x ^ { 5 } + 5 x$ 
B)  $P ( x ) = 3 x ^ { 5 } + 94$ 
C)  $P ( x ) = 3 x ^ { 5 } - 5 x + 94$ 
D)  $P ( x ) = 60 x ^ { 3 } - 5 x + 94$ 
E)  $P ( x ) = 3 x ^ { 5 } + C$ 
A)  $P ( x ) = 3 x ^ { 5 } + 5 x$ 
B)  $P ( x ) = 3 x ^ { 5 } + 94$ 
C)  $P ( x ) = 3 x ^ { 5 } - 5 x + 94$ 
D)  $P ( x ) = 60 x ^ { 3 } - 5 x + 94$ 
E)  $P ( x ) = 3 x ^ { 5 } + C$

Question 11

Use the error formula to estimate the error in approximating the integral $\int _ { 0 } ^ { \pi } 3 \cos x d x$ with $n = 6$ using Simpson's Rule. Round your answer to six decimal places.

Accepted Answer

A)  $0.007084$ 
B)  $0.141676$ 
C)  $0.004723$ 
D)  $0.003936$ 
E)  $0.002530$ 
A)  $0.007084$ 
B)  $0.141676$ 
C)  $0.004723$ 
D)  $0.003936$ 
E)  $0.002530$

Question 12

Find the indefinite integral $\int t ^ { 3 } \left( 5 + t ^ { 4 } \right) ^ { 5 } d t$

Accepted Answer

A)  $\frac { \left( 5 + t ^ { 3 } \right) ^ { 6 } } { 24 } + C$ 
B)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 24 } + C$ 
C)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 6 } + C$ 
D)  $24 \left( 5 + t ^ { 4 } \right) ^ { 6 } + C$ 
E)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 16 } + C$ 
A)  $\frac { \left( 5 + t ^ { 3 } \right) ^ { 6 } } { 24 } + C$ 
B)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 24 } + C$ 
C)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 6 } + C$ 
D)  $24 \left( 5 + t ^ { 4 } \right) ^ { 6 } + C$ 
E)  $\frac { \left( 5 + t ^ { 4 } \right) ^ { 6 } } { 16 } + C$

Question 13

Evaluate the definite integral of the trig function. $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$ Use a graphing utility to verify your results.

Accepted Answer

A)  $- 7$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 8$ 
E)  $- 4$ 
A)  $- 7$ 
B)  $- 5$ 
C)  $- 6$ 
D)  $- 8$ 
E)  $- 4$

Question 14

$\text { Find } \int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x \text {. }$

Accepted Answer

A)  $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } - 26 x + 322 \ln | x + 12 | + C$ 
B) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } - 26 x + 34 \ln | x + 12 | + C$ 
C) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = x ^ { 2 } - 26 x + 322 \ln | x + 12 | + C$ 
D) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } + 2 x + 34 \ln | x + 12 | + C$ 
E) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = x ^ { 2 } + 2 x + 34 \ln | x + 12 | + C$ 
A)  $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } - 26 x + 322 \ln | x + 12 | + C$ 
B) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } - 26 x + 34 \ln | x + 12 | + C$ 
C) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = x ^ { 2 } - 26 x + 322 \ln | x + 12 | + C$ 
D) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = \frac { 1 } { 2 } x ^ { 2 } + 2 x + 34 \ln | x + 12 | + C$ 
E) $\int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x = x ^ { 2 } + 2 x + 34 \ln | x + 12 | + C$

Question 15

Find the sum given below. $$\sum _ { k = 3 } ^ { 8 } k ( k - 4 )$$

Accepted Answer

A)  3 
B)  33 
C)  25 
D)  67 
E)  43 
A)  3 
B)  33 
C)  25 
D)  67 
E)  43

Question 16

Evaluate the definite integral of the function. 
$\int _ { 0 } ^ { 5 x } ( 2 t + 5 \cos t ) d t$ Use a graphing utility to verify your results.

Accepted Answer

A)  $50 \pi ^ { 2 }$ 
B)  $25 \pi ^ { 2 } + 10$ 
C)  $25 \pi ^ { 2 }$ 
D)  $25 \pi ^ { 2 } - 10$ 
E)  $5 \pi ^ { 2 }$ 
A)  $50 \pi ^ { 2 }$ 
B)  $25 \pi ^ { 2 } + 10$ 
C)  $25 \pi ^ { 2 }$ 
D)  $25 \pi ^ { 2 } - 10$ 
E)  $5 \pi ^ { 2 }$

Question 17

Find the indefinite integral $$\int 2 \sec y ( \tan y - \sec y ) d y .$$

Accepted Answer

A)  $2 \sec y \tan y + C$ 
B)  $2 \sec y - 2 \tan y + C$ 
C)  $2 \sec y + \tan y + C$ 
D)  $2 + 2 \sec y + C$ 
E)  2 secy tany $- 2 \sec y + C$ 
A)  $2 \sec y \tan y + C$ 
B)  $2 \sec y - 2 \tan y + C$ 
C)  $2 \sec y + \tan y + C$ 
D)  $2 + 2 \sec y + C$ 
E)  2 secy tany $- 2 \sec y + C$

Question 18

Find the indefinite integral. $$\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$$

Accepted Answer

A)  $\arcsin \left( \frac { x + 18 } { 18 } \right) + C$ 
B)  $\arcsin \left( \frac { x + 9 } { 9 } \right) + C$ 
C)  $\arcsin \left( \frac { x - 18 } { 18 } \right) + C$ 
D)  $\arcsin \left( \frac { x - 9 } { 9 } \right) + C$ 
E)  $\arcsin \left( \frac { x + 9 } { 18 } \right) + C$ 
A)  $\arcsin \left( \frac { x + 18 } { 18 } \right) + C$ 
B)  $\arcsin \left( \frac { x + 9 } { 9 } \right) + C$ 
C)  $\arcsin \left( \frac { x - 18 } { 18 } \right) + C$ 
D)  $\arcsin \left( \frac { x - 9 } { 9 } \right) + C$ 
E)  $\arcsin \left( \frac { x + 9 } { 18 } \right) + C$

Question 19

$\text { Find the indefinite integral of } \int \frac { 1 } { x + 5 } d x \text {. }$

Accepted Answer

A)  $- \ln | x + 5 | + C$ 
B)  $\frac { 1 } { x + 5 } + C$ 
C)  $\frac { 1 } { ( x + 5 ) ^ { 2 } } + C$ 
D)  $\frac { 1 } { x ^ { 2 } + 5 } + C$ 
E)  $\ln | x + 5 | + C$ 
A)  $- \ln | x + 5 | + C$ 
B)  $\frac { 1 } { x + 5 } + C$ 
C)  $\frac { 1 } { ( x + 5 ) ^ { 2 } } + C$ 
D)  $\frac { 1 } { x ^ { 2 } + 5 } + C$ 
E)  $\ln | x + 5 | + C$

Question 20

$\text { The rate of change in sales } S \text { is inversely proportional to time } t ( t > 1 ) \text { measured in }$ weeks. Find $S$ as a function of $t$ if sales after 2 and 4 weeks are 200 units and 350 units respectively.

Accepted Answer

A)  $S ( t ) = 50 \left[ \frac { 11 \ln t } { \ln 2 } + 1 \right]$ 
B)  $S ( t ) = 550 \left[ \frac { \ln t } { \ln 3 } + 1 \right]$ 
C)  $S ( t ) = 550 \left[ \frac { \ln t } { \ln 3 } + 150 \right]$ 
D)  $S ( t ) = 50 \left[ \frac { 3 \ln t } { \ln 2 } + 1 \right]$ 
E)  $S ( t ) = 50 \left[ \frac { 3 \ln t } { \ln 2 } + 11 \right]$ 
A)  $S ( t ) = 50 \left[ \frac { 11 \ln t } { \ln 2 } + 1 \right]$ 
B)  $S ( t ) = 550 \left[ \frac { \ln t } { \ln 3 } + 1 \right]$ 
C)  $S ( t ) = 550 \left[ \frac { \ln t } { \ln 3 } + 150 \right]$ 
D)  $S ( t ) = 50 \left[ \frac { 3 \ln t } { \ln 2 } + 1 \right]$ 
E)  $S ( t ) = 50 \left[ \frac { 3 \ln t } { \ln 2 } + 11 \right]$

Evaluate the following definite integral. $\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 6 t + 3 } } d t$ Use a graphing utility to check your answer.

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. $\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x$

Use the Trapezoidal Rule to approximate the value of the definite integral $\int_{\frac{\pi}{2}}^{x} 5 \sqrt{x} \sin 4 x d x$ with $n = 4$ . Round your answer to four decimal places. $\frac { x } { 2 }$

Find the area of the shaded region. $y = \frac { 1 } { 4 } e ^ { x }$ $Find the area of the shaded region. y = \frac { 1 } { 4 } e ^ { x }$

$\text { Use the differential equation } \frac { d y } { d x } = \frac { 6 } { \sqrt { 64 - x ^ { 2 } } } \text { and the initial condition } y ( 0 ) = \pi \text { to }$ find y.

$\text { Use integration to find the particular solution of the differential equation } \frac { d y } { d x } = \frac { \ln x } { x }$ which passes through the point $( 1 , - 3 )$ .

Evaluate the definite integral of the algebraic function. $\int _ { 0 } ^ { 5 } 4 | x - 4 | d x$ Use a graphing utility to verify your results.

$\text { Find } F ^ {t } ( x ) \text { if } F ( x ) = \int _ { 1 } ^ { 14 x ^ { 2 } } \frac { 1 } { t } d t$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 34 } ( 4 i - 8 )$

Solve the differential equation. $\frac { d P } { d x } = 15 x ^ { 4 } - 5 , \quad P ( - 2 ) = 8$

Use the error formula to estimate the error in approximating the integral $\int _ { 0 } ^ { \pi } 3 \cos x d x$ with $n = 6$ using Simpson's Rule. Round your answer to six decimal places.

Find the indefinite integral $\int t ^ { 3 } \left( 5 + t ^ { 4 } \right) ^ { 5 } d t$

Evaluate the definite integral of the trig function. $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$ Use a graphing utility to verify your results.

$\text { Find } \int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x \text {. }$

Find the sum given below. $\sum _ { k = 3 } ^ { 8 } k ( k - 4 )$

Evaluate the definite integral of the function. $\int _ { 0 } ^ { 5 x } ( 2 t + 5 \cos t ) d t$ Use a graphing utility to verify your results.

Find the indefinite integral $\int 2 \sec y ( \tan y - \sec y ) d y .$

Find the indefinite integral. $\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$

$\text { Find the indefinite integral of } \int \frac { 1 } { x + 5 } d x \text {. }$

$\text { The rate of change in sales } S \text { is inversely proportional to time } t ( t > 1 ) \text { measured in }$ weeks. Find $S$ as a function of $t$ if sales after 2 and 4 weeks are 200 units and 350 units respectively.

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

Evaluate the following definite integral. ∫3516t+3dt\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 6 t + 3 } } d t∫35​6t+3​1​dt Use a graphing utility to check your answer.

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. ∫34x4dx\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x∫34​4x​dx

Use the Trapezoidal Rule to approximate the value of the definite integral ∫π2x5xsin⁡4xdx \int_{\frac{\pi}{2}}^{x} 5 \sqrt{x} \sin 4 x d x ∫2π​x​5x​sin4xdx with n=4n = 4n=4 . Round your answer to four decimal places. x2\frac { x } { 2 }2x​

Find the area of the shaded region. y=14exy = \frac { 1 } { 4 } e ^ { x }y=41​ex

Evaluate the definite integral of the algebraic function. ∫054∣x−4∣dx\int _ { 0 } ^ { 5 } 4 | x - 4 | d x∫05​4∣x−4∣dx Use a graphing utility to verify your results.

Find Ft(x) if F(x)=∫114x21tdt\text { Find } F ^ {t } ( x ) \text { if } F ( x ) = \int _ { 1 } ^ { 14 x ^ { 2 } } \frac { 1 } { t } d t Find Ft(x) if F(x)=∫114x2​t1​dt

Use the properties of summation and Theorem 5.2 to evaluate the sum. ∑i=134(4i−8)\sum _ { i = 1 } ^ { 34 } ( 4 i - 8 )∑i=134​(4i−8)

Solve the differential equation. dPdx=15x4−5,P(−2)=8\frac { d P } { d x } = 15 x ^ { 4 } - 5 , \quad P ( - 2 ) = 8dxdP​=15x4−5,P(−2)=8

Use the error formula to estimate the error in approximating the integral ∫0π3cos⁡xdx\int _ { 0 } ^ { \pi } 3 \cos x d x∫0π​3cosxdx with n=6n = 6n=6 using Simpson's Rule. Round your answer to six decimal places.

Find the indefinite integral ∫t3(5+t4)5dt\int t ^ { 3 } \left( 5 + t ^ { 4 } \right) ^ { 5 } d t∫t3(5+t4)5dt

Evaluate the definite integral of the trig function. ∫05π(−3sin⁡x+4cos⁡x)dx\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x∫05π​(−3sinx+4cosx)dx Use a graphing utility to verify your results.

Find ∫x2−14x+10x+12dx. \text { Find } \int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x \text {. } Find ∫x+12x2−14x+10​dx.

Find the sum given below. ∑k=38k(k−4)\sum _ { k = 3 } ^ { 8 } k ( k - 4 )∑k=38​k(k−4)

Evaluate the definite integral of the function. ∫05x(2t+5cos⁡t)dt\int _ { 0 } ^ { 5 x } ( 2 t + 5 \cos t ) d t∫05x​(2t+5cost)dt Use a graphing utility to verify your results.

Find the indefinite integral ∫2sec⁡y(tan⁡y−sec⁡y)dy.\int 2 \sec y ( \tan y - \sec y ) d y .∫2secy(tany−secy)dy.

Find the indefinite integral. ∫dx−x2−18x\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }∫−x2−18x​dx​

Find the indefinite integral of ∫1x+5dx. \text { Find the indefinite integral of } \int \frac { 1 } { x + 5 } d x \text {. } Find the indefinite integral of ∫x+51​dx.

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

Evaluate the following definite integral. $\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 6 t + 3 } } d t$ Use a graphing utility to check your answer.

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. $\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x$

Use the Trapezoidal Rule to approximate the value of the definite integral $\int_{\frac{\pi}{2}}^{x} 5 \sqrt{x} \sin 4 x d x$ with $n = 4$ . Round your answer to four decimal places. $\frac { x } { 2 }$

Find the area of the shaded region. $y = \frac { 1 } { 4 } e ^ { x }$ $Find the area of the shaded region. y = \frac { 1 } { 4 } e ^ { x }$

Evaluate the definite integral of the algebraic function. $\int _ { 0 } ^ { 5 } 4 | x - 4 | d x$ Use a graphing utility to verify your results.

$\text { Find } F ^ {t } ( x ) \text { if } F ( x ) = \int _ { 1 } ^ { 14 x ^ { 2 } } \frac { 1 } { t } d t$

Use the properties of summation and Theorem 5.2 to evaluate the sum. $\sum _ { i = 1 } ^ { 34 } ( 4 i - 8 )$

Solve the differential equation. $\frac { d P } { d x } = 15 x ^ { 4 } - 5 , \quad P ( - 2 ) = 8$

Use the error formula to estimate the error in approximating the integral $\int _ { 0 } ^ { \pi } 3 \cos x d x$ with $n = 6$ using Simpson's Rule. Round your answer to six decimal places.

Find the indefinite integral $\int t ^ { 3 } \left( 5 + t ^ { 4 } \right) ^ { 5 } d t$

Evaluate the definite integral of the trig function. $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$ Use a graphing utility to verify your results.

$\text { Find } \int \frac { x ^ { 2 } - 14 x + 10 } { x + 12 } d x \text {. }$

Find the sum given below. $\sum _ { k = 3 } ^ { 8 } k ( k - 4 )$

Evaluate the definite integral of the function. $\int _ { 0 } ^ { 5 x } ( 2 t + 5 \cos t ) d t$ Use a graphing utility to verify your results.

Find the indefinite integral $\int 2 \sec y ( \tan y - \sec y ) d y .$

Find the indefinite integral. $\int \frac { d x } { \sqrt { - x ^ { 2 } - 18 x } }$

$\text { Find the indefinite integral of } \int \frac { 1 } { x + 5 } d x \text {. }$