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[T] 2 e −2 x 1 e −4 x d x over [ 0 , 2 ]

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[T] 1 x + x ln 2 x over [ 0 , 2 ]


A graph of the function f(x) = arctan(ln(x)) over (0, 2]. It is an increasing curve with x-intercept at (1,0).
The general antiderivative is tan −1 ( ln x ) + C . Taking C = π 2 = tan −1 recovers the definite integral.

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[T] sin −1 x 1 x 2 over [ −1 , 1 ]

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In the following exercises, compute each integral using appropriate substitutions.

e x 1 e 2 t d t

sin −1 ( e t ) + C

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d t t 1 ln 2 t

sin −1 ( ln t ) + C

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d t t ( 1 + ln 2 t )

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cos −1 ( 2 t ) 1 4 t 2 d t

1 2 ( cos −1 ( 2 t ) ) 2 + C

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e t cos −1 ( e t ) 1 e 2 t d t

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In the following exercises, compute each definite integral.

0 1 / 2 tan ( sin −1 t ) 1 t 2 d t

1 2 ln ( 4 3 )

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1 / 4 1 / 2 tan ( cos −1 t ) 1 t 2 d t

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0 1 / 2 sin ( tan −1 t ) 1 + t 2 d t

1 2 5

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0 1 / 2 cos ( tan −1 t ) 1 + t 2 d t

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For A > 0 , compute I ( A ) = A A d t 1 + t 2 and evaluate lim a I ( A ) , the area under the graph of 1 1 + t 2 on [ , ] .

2 tan −1 ( A ) π as A

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For 1 < B < , compute I ( B ) = 1 B d t t t 2 1 and evaluate lim B I ( B ) , the area under the graph of 1 t t 2 1 over [ 1 , ) .

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Use the substitution u = 2 cot x and the identity 1 + cot 2 x = csc 2 x to evaluate d x 1 + cos 2 x . ( Hint: Multiply the top and bottom of the integrand by csc 2 x . )

Using the hint, one has csc 2 x csc 2 x + cot 2 x d x = csc 2 x 1 + 2 cot 2 x d x . Set u = 2 cot x . Then, d u = 2 csc 2 x and the integral is 1 2 d u 1 + u 2 = 1 2 tan −1 u + C = 1 2 tan −1 ( 2 cot x ) + C . If one uses the identity tan −1 s + tan −1 ( 1 s ) = π 2 , then this can also be written 1 2 tan −1 ( tan x 2 ) + C .

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[T] Approximate the points at which the graphs of f ( x ) = 2 x 2 1 and g ( x ) = ( 1 + 4 x 2 ) −3 / 2 intersect, and approximate the area between their graphs accurate to three decimal places.

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47. [T] Approximate the points at which the graphs of f ( x ) = x 2 1 and f ( x ) = x 2 1 intersect, and approximate the area between their graphs accurate to three decimal places.

x ± 1.13525 . The left endpoint estimate with N = 100 is 2.796 and these decimals persist for N = 500 .

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Use the following graph to prove that 0 x 1 t 2 d t = 1 2 x 1 x 2 + 1 2 sin −1 x .


A diagram containing two shapes, a wedge from a circle shaded in blue on top of a triangle shaded in brown. The triangle’s hypotenuse is one of the radii edges of the wedge of the circle and is 1 unit long. There is a dotted red line forming a rectangle out of part of the wedge and the triangle, with the hypotenuse of the triangle as the diagonal of the rectangle. The curve of the circle is described by the equation sqrt(1-x^2).

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Chapter review exercises

True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous over their domains.

If f ( x ) > 0 , f ( x ) > 0 for all x , then the right-hand rule underestimates the integral a b f ( x ) . Use a graph to justify your answer.

False

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a b f ( x ) 2 d x = a b f ( x ) d x a b f ( x ) d x

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If f ( x ) g ( x ) for all x [ a , b ] , then a b f ( x ) a b g ( x ) .

True

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All continuous functions have an antiderivative.

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Evaluate the Riemann sums L 4 and R 4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.

y = 3 x 2 2 x + 1 over [ −1 , 1 ]

L 4 = 5.25 , R 4 = 3.25 , exact answer: 4

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y = ln ( x 2 + 1 ) over [ 0 , e ]

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y = x 2 sin x over [ 0 , π ]

L 4 = 5.364 , R 4 = 5.364 , exact answer: 5.870

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y = x + 1 x over [ 1 , 4 ]

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Evaluate the following integrals.

−1 1 ( x 3 2 x 2 + 4 x ) d x

4 3

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0 4 3 t 1 + 6 t 2 d t

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π / 3 π / 2 2 sec ( 2 θ ) tan ( 2 θ ) d θ

1

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0 π / 4 e cos 2 x sin x cos d x

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Find the antiderivative.

d x ( x + 4 ) 3

1 2 ( x + 4 ) 2 + C

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4 x 2 1 x 6 d x

4 3 sin −1 ( x 3 ) + C

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Find the derivative.

d d t 0 t sin x 1 + x 2 d x

sin t 1 + t 2

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d d x 1 x 3 4 t 2 d t

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d d x 1 ln ( x ) ( 4 t + e t ) d t

4 ln x x + 1

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d d x 0 cos x e t 2 d t

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The following problems consider the historic average cost per gigabyte of RAM on a computer.

Year 5-Year Change ($)
1980 0
1985 −5,468,750
1990 755,495
1995 −73,005
2000 −29,768
2005 −918
2010 −177

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

$6,328,113

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The average cost per gigabyte of RAM can be approximated by the function C ( t ) = 8 , 500 , 000 ( 0.65 ) t , where t is measured in years since 1980, and C is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

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Find the average cost of 1GB RAM for 2005 to 2010.

$73.36

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The velocity of a bullet from a rifle can be approximated by v ( t ) = 6400 t 2 6505 t + 2686 , where t is seconds after the shot and v is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: 0 t 0.5 . What is the total distance the bullet travels in 0.5 sec?

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What is the average velocity of the bullet for the first half-second?

19117 12 ft/sec , or 1593 ft/sec

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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