Question 1

Find the indefinite integral. $\int e ^ { - 4 y } d y$

Accepted Answer

A)  $- 4 e ^ { - 4 y } + C$ 
B)  $- 4 e ^ { - 5 y } + C$ 
C)  $- \frac { 1 } { 4 } e ^ { - 5 y } + C$ 
D)  $- \frac { 1 } { 4 } e ^ { - 4 y } + C$ 
E)  $- \frac { 1 } { 4 } e ^ { \frac { - 1 } { 4 } y } + C$ 
A)  $- 4 e ^ { - 4 y } + C$ 
B)  $- 4 e ^ { - 5 y } + C$ 
C)  $- \frac { 1 } { 4 } e ^ { - 5 y } + C$ 
D)  $- \frac { 1 } { 4 } e ^ { - 4 y } + C$ 
E)  $- \frac { 1 } { 4 } e ^ { \frac { - 1 } { 4 } y } + C$

Question 2

Find the area of the shaded region.

Accepted Answer

A)  $\frac { 13 } { 6 }$ 
B)  $\frac { 37 } { 12 }$ 
C)  $\frac { 37 } { 6 }$ 
D)  $\frac { 13 } { 12 }$ 
E)  $\frac { 13 } { 7 }$ 
A)  $\frac { 13 } { 6 }$ 
B)  $\frac { 37 } { 12 }$ 
C)  $\frac { 37 } { 6 }$ 
D)  $\frac { 13 } { 12 }$ 
E)  $\frac { 13 } { 7 }$

Question 3

The integrand of the following definite integral is a difference of two functions. $$\int _ { 0 } ^ { 4 } \left[ ( x + 1 ) - \frac { 1 } { 2 } x \right] d x$$ Sketch the graph of the two functions and shade the region whose area is represented by the integral.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 4

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$

Accepted Answer

A)  $2 \sqrt { t ^ { 3 } + 3 } + C$ 
B)  $\sqrt { t ^ { 3 } + 3 } + C$ 
C)  $2 \sqrt { t ^ { 2 } + 3 } + C$ 
D)  $\frac { 1 } { 2 } \sqrt { t ^ { 3 } + 3 } + C$ 
E)  $\frac { 1 } { 2 } \sqrt { t ^ { 2 } + 3 } + C$ 
A)  $2 \sqrt { t ^ { 3 } + 3 } + C$ 
B)  $\sqrt { t ^ { 3 } + 3 } + C$ 
C)  $2 \sqrt { t ^ { 2 } + 3 } + C$ 
D)  $\frac { 1 } { 2 } \sqrt { t ^ { 3 } + 3 } + C$ 
E)  $\frac { 1 } { 2 } \sqrt { t ^ { 2 } + 3 } + C$

Question 5

The rate of depreciation of a building is given by $D ^ { \prime } ( t ) = 8,300 ( 20 - t )$ dollars per year, $0 \leq t \leq 20$ Use the definite integral to find the total depreciation over the first $20$ years.

Accepted Answer

A)  $\$ 1,660,000$ 
B)  $\$ 83,000$ 
C)  $\$ 830,000$ 
D)  $\$ 474,286$ 
E)  $\$ 3,320,000$ 
A)  $\$ 1,660,000$ 
B)  $\$ 83,000$ 
C)  $\$ 830,000$ 
D)  $\$ 474,286$ 
E)  $\$ 3,320,000$

Question 6

Evaluate the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$

Accepted Answer

A)  $\frac { 1 } { 140 } \left( 4 x ^ { 5 } + 3 \right) ^ { 7 } + C$ 
B)  $\frac { 20 } { 9 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$ 
C)  $\frac { 5 } { 2 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$ 
D)  $\frac { 1 } { 180 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$ 
E)  $\frac { 1 } { 160 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$ 
A)  $\frac { 1 } { 140 } \left( 4 x ^ { 5 } + 3 \right) ^ { 7 } + C$ 
B)  $\frac { 20 } { 9 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$ 
C)  $\frac { 5 } { 2 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$ 
D)  $\frac { 1 } { 180 } \left( 4 x ^ { 5 } + 3 \right) ^ { 9 } + C$ 
E)  $\frac { 1 } { 160 } \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } + C$

Question 7

Evaluate the integral $\int e ^ { 23 x } d x$

Accepted Answer

A)  $\frac { 1 } { 23 } e ^ { 23 x } + C$ 
B)  $23 e ^ {2 3 x } + C$ 
C)  $\frac { 1 } { 24 } e ^ { 24 x } + C$ 
D)  $23 e ^ { 22 x } + C$ 
E)  $\frac { 1 } { 22 } e ^ { 2 2 x } + C$ 
A)  $\frac { 1 } { 23 } e ^ { 23 x } + C$ 
B)  $23 e ^ {2 3 x } + C$ 
C)  $\frac { 1 } { 24 } e ^ { 24 x } + C$ 
D)  $23 e ^ { 22 x } + C$ 
E)  $\frac { 1 } { 22 } e ^ { 2 2 x } + C$

Question 8

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 1 + 4 z ) ^ { 3 } d z$

Accepted Answer

A)  $4 ( 1 + 4 z ) ^ { 4 } + C$ 
B)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 3 } + C$ 
C)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 4 } + C$ 
D)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 16 } + C$ 
E) none of the above 
A)  $4 ( 1 + 4 z ) ^ { 4 } + C$ 
B)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 3 } + C$ 
C)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 4 } + C$ 
D)  $\frac { ( 1 + 4 z ) ^ { 4 } } { 16 } + C$ 
E) none of the above

Question 9

Find the indefinite integral. $\int \frac { ( \ln x ) ^ { 4 } } { x } d x$

Accepted Answer

A)  $\frac { ( \ln x ) ^ { 4 } } { 4 } + C$ 
B)  $4 ( \ln x ) ^ { 3 } + C$ 
C)  $\frac { 5 \ln x } { x } + C$ 
D)  $\frac { ( \ln x ) ^ { 5 } } { 5 } + C$ 
E) none of the above 
A)  $\frac { ( \ln x ) ^ { 4 } } { 4 } + C$ 
B)  $4 ( \ln x ) ^ { 3 } + C$ 
C)  $\frac { 5 \ln x } { x } + C$ 
D)  $\frac { ( \ln x ) ^ { 5 } } { 5 } + C$ 
E) none of the above

Question 10

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

Accepted Answer

A)  $12 \left( 5 + u ^ { 3 } \right) ^ { 4 } + C$ 
B)  $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 12 } + C$ 
C)  $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 4 } + C$ 
D)  $\frac { \left( 5 + u ^ { 2 } \right) ^ { 4 } } { 12 } + C$ 
E) none of the above 
A)  $12 \left( 5 + u ^ { 3 } \right) ^ { 4 } + C$ 
B)  $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 12 } + C$ 
C)  $\frac { \left( 5 + u ^ { 3 } \right) ^ { 4 } } { 4 } + C$ 
D)  $\frac { \left( 5 + u ^ { 2 } \right) ^ { 4 } } { 12 } + C$ 
E) none of the above

Question 11

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function$~~~~~~~~~~~~~~~~$ Supply Function
$P = 975 - 23 x ~~~~~~~~~~~\quad 42 x$

Accepted Answer

A) a. $2587.50b. $3725.00 
B) a. $5587.50b. $4725.00 
C) a. $2587.50b. $1725.00 
D) a. $1587.50b. $4725.00 
E) a. $3587.50b. $4725.00 
A) a. $2587.50b. $3725.00 
B) a. $5587.50b. $4725.00 
C) a. $2587.50b. $1725.00 
D) a. $1587.50b. $4725.00 
E) a. $3587.50b. $4725.00

Question 12

Use formal substitution to find the indefinite integral $\int \frac { x ^ { 4 } + 1 } { \sqrt { x ^ { 5 } + 5 x + 6 } } d x$ .

Accepted Answer

A)  $\frac { 2 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$ 
B)  $\frac { 1 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$ 
C)  $\frac { 2 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$ 
D)  $\frac { 2 } { 5 } \sqrt { x ^ { 5 } + 5 x + 6 } + C$ 
E)  $\frac { 1 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$ 
A)  $\frac { 2 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$ 
B)  $\frac { 1 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$ 
C)  $\frac { 2 } { 5 } \sqrt { x ^ { 4 } + 5 x + 6 } + C$ 
D)  $\frac { 2 } { 5 } \sqrt { x ^ { 5 } + 5 x + 6 } + C$ 
E)  $\frac { 1 } { 5 } \left( x ^ { 5 } + 5 x + 6 \right) + C$

Question 13

Find the particular solution that satisfies the differential equation $f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - 5$ and initial condition $f ( 4 ) = - 16$ .

Accepted Answer

A)  $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } - 5 x$ 
B)  $f ( x ) = \frac { 1 } { 7 } x ^ { 2 } + 5 x - 1260$ 
C)  $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x$ 
D)  $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x - 1260$ 
E)  $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } + 5 x$ 
A)  $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } - 5 x$ 
B)  $f ( x ) = \frac { 1 } { 7 } x ^ { 2 } + 5 x - 1260$ 
C)  $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x$ 
D)  $f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - 5 x - 1260$ 
E)  $f ( x ) = \frac { 1 } { 5 } x ^ { 2 } + 5 x$

Question 14

Use the Midpoint Rule with $n = 4$ to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. $f ( x ) = \left( x ^ { 2 } - 4 \right) ^ { 2 } , [ - 2,2 ]$

Accepted Answer

A) The approximate area is: $\approx 30.25$   
B) The approximate area is: $\approx 24.25$   
C) The approximate area is: $\approx 34.25$   
D) The approximate area is: $\approx 14.25$   
E) The approximate area is: $\approx 34.99$   
A) The approximate area is: $\approx 30.25$   
B) The approximate area is: $\approx 24.25$   
C) The approximate area is: $\approx 34.25$   
D) The approximate area is: $\approx 14.25$   
E) The approximate area is: $\approx 34.99$

Question 15

The marginal cost of a product is modeled by $\frac { d C } { d x } = \frac { 8 } { \sqrt { x + 1 } }$ , when x = 8, C = 40. Find the cost function.

Accepted Answer

A)  $C ( x ) = 8 \sqrt { x + 1 } + - 8$ 
B)  $C ( x ) = 16 ( x + 1 ) + - 6$ 
C)  $C ( x ) = 16 \sqrt { x + 1 } + - 8$ 
D)  $C ( x ) = 8 \sqrt { x + 1 } + - 8$ 
E)  $C ( x ) = 8 ( x + 1 ) + - 6$ 
A)  $C ( x ) = 8 \sqrt { x + 1 } + - 8$ 
B)  $C ( x ) = 16 ( x + 1 ) + - 6$ 
C)  $C ( x ) = 16 \sqrt { x + 1 } + - 8$ 
D)  $C ( x ) = 8 \sqrt { x + 1 } + - 8$ 
E)  $C ( x ) = 8 ( x + 1 ) + - 6$

Question 16

Find a function that satisfies the conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$ .

Accepted Answer

A)  $f ( x ) = \frac { 1 } { 6 } x ^ { 6 } + 9 x$ 
B)  $f ( x ) = \frac { 1 } { 42 } x ^ { 7 } + 9 x + 5$ 
C)  $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 9 x + 5$ 
D)  $f ( x ) = \frac { 1 } { 7 } x ^ { 7 } + 9 x$ 
E)  $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 5 x + 9$ 
A)  $f ( x ) = \frac { 1 } { 6 } x ^ { 6 } + 9 x$ 
B)  $f ( x ) = \frac { 1 } { 42 } x ^ { 7 } + 9 x + 5$ 
C)  $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 9 x + 5$ 
D)  $f ( x ) = \frac { 1 } { 7 } x ^ { 7 } + 9 x$ 
E)  $f ( x ) = \frac { 1 } { 42 } x ^ { 6 } + 5 x + 9$

Question 17

Find the indefinite integral. $\int \frac { x ^ { 2 } } { 3 x ^ { 3 } + 10 } d x$

Accepted Answer

A)  $\frac { 1 } { 9 } \ln \left| 3 x ^ { 3 } + 10 \right| + C$ 
B)  $\ln \left| 3 x ^ { 3 } + 10 \right| + C$ 
C)  $\frac { x ^ { 3 } } { 3 x ^ { 4 } + 10 x } + C$ 
D) integral does not exist 
E) none of the above 
A)  $\frac { 1 } { 9 } \ln \left| 3 x ^ { 3 } + 10 \right| + C$ 
B)  $\ln \left| 3 x ^ { 3 } + 10 \right| + C$ 
C)  $\frac { x ^ { 3 } } { 3 x ^ { 4 } + 10 x } + C$ 
D) integral does not exist 
E) none of the above

Question 18

Evaluate the definite integral $\int _ { 4 } ^ { 7 } ( x - 7 ) ^ { 5 } d x$ .

Accepted Answer

A)  $\frac { 729 } { 5 }$ 
B)  $- \frac { 729 } { 7 }$ 
C)  $- \frac { 729 } { 5 }$ 
D)  $\frac { 243 } { 2 }$ 
E)  $- \frac { 243 } { 2 }$ 
A)  $\frac { 729 } { 5 }$ 
B)  $- \frac { 729 } { 7 }$ 
C)  $- \frac { 729 } { 5 }$ 
D)  $\frac { 243 } { 2 }$ 
E)  $- \frac { 243 } { 2 }$

Question 19

Evaluate the integral $\int x ^ { 7 } e ^ { 7 x ^ { 8 } } d x$

Accepted Answer

A)  $\frac { 1 } { 8 } e ^ { 7 x ^ { 8 } } + C$ 
B)  $56 e ^ { 7 x ^ { 8 } } + C$ 
C)  $\frac { 1 } { 56 } e ^ { 7 x ^ { 8 } } + C$ 
D)  $8 e ^ { 7 x ^ { 8 } } + C$ 
E)  $\frac { 1 } { 56 } x ^ { 8 } e ^ { 7 x ^ { 8 } } + C$ 
A)  $\frac { 1 } { 8 } e ^ { 7 x ^ { 8 } } + C$ 
B)  $56 e ^ { 7 x ^ { 8 } } + C$ 
C)  $\frac { 1 } { 56 } e ^ { 7 x ^ { 8 } } + C$ 
D)  $8 e ^ { 7 x ^ { 8 } } + C$ 
E)  $\frac { 1 } { 56 } x ^ { 8 } e ^ { 7 x ^ { 8 } } + C$

Question 20

Find the average value of the function over the given interval. $f ( x ) = 9 - x ^ { 2 }$ on [-3,3]

Accepted Answer

A) 6 
B) 21 
C) 36 
D) 4 
E) 20 
A) 6 
B) 21 
C) 36 
D) 4 
E) 20

Find the indefinite integral. $\int e ^ { - 4 y } d y$

Find the area of the shaded region.

The integrand of the following definite integral is a difference of two functions. $\int _ { 0 } ^ { 4 } \left[ ( x + 1 ) - \frac { 1 } { 2 } x \right] d x$ Sketch the graph of the two functions and shade the region whose area is represented by the integral.

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$

The rate of depreciation of a building is given by $D ^ { \prime } ( t ) = 8,300 ( 20 - t )$ dollars per year, $0 \leq t \leq 20$ Use the definite integral to find the total depreciation over the first $20$ years.

Evaluate the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$

Evaluate the integral $\int e ^ { 23 x } d x$

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 1 + 4 z ) ^ { 3 } d z$

Find the indefinite integral. $\int \frac { ( \ln x ) ^ { 4 } } { x } d x$

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

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function $~~~~~~$ Supply Function $P = 975 - 23 x ~\quad 42 x$

Use formal substitution to find the indefinite integral $\int \frac { x ^ { 4 } + 1 } { \sqrt { x ^ { 5 } + 5 x + 6 } } d x$ .

Find the particular solution that satisfies the differential equation $f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - 5$ and initial condition $f ( 4 ) = - 16$ .

Use the Midpoint Rule with $n = 4$ to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. $f ( x ) = \left( x ^ { 2 } - 4 \right) ^ { 2 } , [ - 2,2 ]$

The marginal cost of a product is modeled by $\frac { d C } { d x } = \frac { 8 } { \sqrt { x + 1 } }$ , when x = 8, C = 40. Find the cost function.

Find a function that satisfies the conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$ .

Find the indefinite integral. $\int \frac { x ^ { 2 } } { 3 x ^ { 3 } + 10 } d x$

Evaluate the definite integral $\int _ { 4 } ^ { 7 } ( x - 7 ) ^ { 5 } d x$ .

Evaluate the integral $\int x ^ { 7 } e ^ { 7 x ^ { 8 } } d x$

Find the average value of the function over the given interval. $f ( x ) = 9 - x ^ { 2 }$ on [-3,3]

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Exam 12: Integration and Its Applications

Find the indefinite integral. ∫e−4ydy\int e ^ { - 4 y } d y∫e−4ydy

Find the area of the shaded region.

Find the indefinite integral of the following function and check the result by differentiation. ∫3t2t3+3dt\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t∫t3+3​3t2​dt

The rate of depreciation of a building is given by D′(t)=8,300(20−t)D ^ { \prime } ( t ) = 8,300 ( 20 - t )D′(t)=8,300(20−t) dollars per year, 0≤t≤200 \leq t \leq 200≤t≤20 Use the definite integral to find the total depreciation over the first 202020 years.

Evaluate the integral ∫(4x5+3)8x4dx\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x∫(4x5+3)8x4dx

Evaluate the integral ∫e23xdx\int e ^ { 23 x } d x∫e23xdx

Find the indefinite integral of the following function and check the result by differentiation. ∫(1+4z)3dz\int ( 1 + 4 z ) ^ { 3 } d z∫(1+4z)3dz

Find the indefinite integral. ∫(ln⁡x)4xdx\int \frac { ( \ln x ) ^ { 4 } } { x } d x∫x(lnx)4​dx

Find the indefinite integral of the following function and check the result by differentiation. ∫u2(5+u3)3du\int u ^ { 2 } \left( 5 + u ^ { 3 } \right) ^ { 3 } d u∫u2(5+u3)3du

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function ~~~~~~~~~~~~~~~~ Supply Function P=975−23x 42xP = 975 - 23 x ~~~~~~~~~~~\quad 42 xP=975−23x 42x

Use formal substitution to find the indefinite integral ∫x4+1x5+5x+6dx\int \frac { x ^ { 4 } + 1 } { \sqrt { x ^ { 5 } + 5 x + 6 } } d x∫x5+5x+6​x4+1​dx .

Find the particular solution that satisfies the differential equation f′(x)=12x−5f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - 5f′(x)=21​x−5 and initial condition f(4)=−16f ( 4 ) = - 16f(4)=−16 .

Use the Midpoint Rule with n=4n = 4n=4 to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. f(x)=(x2−4)2,[−2,2]f ( x ) = \left( x ^ { 2 } - 4 \right) ^ { 2 } , [ - 2,2 ]f(x)=(x2−4)2,[−2,2]

The marginal cost of a product is modeled by dCdx=8x+1\frac { d C } { d x } = \frac { 8 } { \sqrt { x + 1 } }dxdC​=x+1​8​ , when x = 8, C = 40. Find the cost function.

Find a function that satisfies the conditions f′′(x)=x5,f′(0)=9,f(0)=5f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5f′′(x)=x5,f′(0)=9,f(0)=5 .

Find the indefinite integral. ∫x23x3+10dx\int \frac { x ^ { 2 } } { 3 x ^ { 3 } + 10 } d x∫3x3+10x2​dx

Evaluate the definite integral ∫47(x−7)5dx\int _ { 4 } ^ { 7 } ( x - 7 ) ^ { 5 } d x∫47​(x−7)5dx .

Evaluate the integral ∫x7e7x8dx\int x ^ { 7 } e ^ { 7 x ^ { 8 } } d x∫x7e7x8dx

Find the average value of the function over the given interval. f(x)=9−x2f ( x ) = 9 - x ^ { 2 }f(x)=9−x2 on [-3,3]

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

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Find the indefinite integral. $\int e ^ { - 4 y } d y$

Find the indefinite integral of the following function and check the result by differentiation. $\int \frac { 3 t ^ { 2 } } { \sqrt { t ^ { 3 } + 3 } } d t$

The rate of depreciation of a building is given by $D ^ { \prime } ( t ) = 8,300 ( 20 - t )$ dollars per year, $0 \leq t \leq 20$ Use the definite integral to find the total depreciation over the first $20$ years.

Evaluate the integral $\int \left( 4 x ^ { 5 } + 3 \right) ^ { 8 } x ^ { 4 } d x$

Evaluate the integral $\int e ^ { 23 x } d x$

Find the indefinite integral of the following function and check the result by differentiation. $\int ( 1 + 4 z ) ^ { 3 } d z$

Find the indefinite integral. $\int \frac { ( \ln x ) ^ { 4 } } { x } d x$

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

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function $~~~~~~$ Supply Function $P = 975 - 23 x ~\quad 42 x$

Use formal substitution to find the indefinite integral $\int \frac { x ^ { 4 } + 1 } { \sqrt { x ^ { 5 } + 5 x + 6 } } d x$ .

Find the particular solution that satisfies the differential equation $f ^ { \prime } ( x ) = \frac { 1 } { 2 } x - 5$ and initial condition $f ( 4 ) = - 16$ .

Use the Midpoint Rule with $n = 4$ to approximate the area of the region bounded by the graph of and the -axis over the interval. Sketch the region. $f ( x ) = \left( x ^ { 2 } - 4 \right) ^ { 2 } , [ - 2,2 ]$

The marginal cost of a product is modeled by $\frac { d C } { d x } = \frac { 8 } { \sqrt { x + 1 } }$ , when x = 8, C = 40. Find the cost function.

Find a function that satisfies the conditions $f ^ { \prime \prime } ( x ) = x ^ { 5 } , f ^ { \prime } ( 0 ) = 9 , f ( 0 ) = 5$ .

Find the indefinite integral. $\int \frac { x ^ { 2 } } { 3 x ^ { 3 } + 10 } d x$

Evaluate the definite integral $\int _ { 4 } ^ { 7 } ( x - 7 ) ^ { 5 } d x$ .

Evaluate the integral $\int x ^ { 7 } e ^ { 7 x ^ { 8 } } d x$

Find the average value of the function over the given interval. $f ( x ) = 9 - x ^ { 2 }$ on [-3,3]