Exam 7: Techniques of Integration

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

(Multiple Choice)

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Question 41

Use the Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$ . Round the result to four decimal places.

(Short Answer)

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Question 42

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

(Multiple Choice)

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Question 43

Evaluate the following integral. $\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$

(Short Answer)

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Question 44

A particle moves on a straight line with velocity function $v ( t ) = \sin a t \cos ^ { 3 } a t$ . Find its position function $s = f ( t )$ if $f ( 0 ) = 0$ .

(Multiple Choice)

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Question 45

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 36 } d x$

(Short Answer)

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Question 46

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

(Multiple Choice)

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Question 47

Evaluate the integral. $\int \frac { \cos ( \ln 18 t ) } { t } d t$

(Short Answer)

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Question 48

Evaluate the integral. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

(Multiple Choice)

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Question 49

Evaluate the integral. $\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$

(Multiple Choice)

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Question 50

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

(Multiple Choice)

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Question 51

$\text { Find the area bounded by the curves } y = 3 \cos x \text { and } y = 3 \cos ^ { 2 } x \text { between } x = 0 \text { and } x = \frac { \pi } { 2 } \text {. }$

(Short Answer)

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Question 52

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

(Multiple Choice)

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Question 53

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 49 } } ; x = 7 \sec \theta$

(Short Answer)

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Question 54

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$

(Multiple Choice)

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Question 55

Evaluate the integral. $\int 4 y ^ { 3 } e ^ { y } d y$

(Short Answer)

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Question 56

Evaluate the integral or show that it is divergent. $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 6 } } d x$

(Short Answer)

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Question 57

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x$

(Short Answer)

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Question 58

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

(Multiple Choice)

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Question 59

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 0 } ^ { \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { 4 - x ^ { 2 } } } d x$

(Multiple Choice)

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Question 60

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Use the Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$ . Round the result to four decimal places.

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Evaluate the following integral. $\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$

A particle moves on a straight line with velocity function $v ( t ) = \sin a t \cos ^ { 3 } a t$ . Find its position function $s = f ( t )$ if $f ( 0 ) = 0$ .

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 36 } d x$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Evaluate the integral. $\int \frac { \cos ( \ln 18 t ) } { t } d t$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral. $\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

$\text { Find the area bounded by the curves } y = 3 \cos x \text { and } y = 3 \cos ^ { 2 } x \text { between } x = 0 \text { and } x = \frac { \pi } { 2 } \text {. }$

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 49 } } ; x = 7 \sec \theta$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$

Evaluate the integral. $\int 4 y ^ { 3 } e ^ { y } d y$

Evaluate the integral or show that it is divergent. $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 6 } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x$

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 0 } ^ { \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { 4 - x ^ { 2 } } } d x$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

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Exam 7: Techniques of Integration

Use long division to evaluate the integral. ∫01x3+12x2−36x+1x2+4x−12dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x∫01​x2+4x−12x3+12x2−36x+1​dx The choices are rounded to 3 decimal places.

Use the Trapezoidal Rule to approximate ∫23e3/xdx\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x∫23​e3/xdx for n=4n = 4n=4 . Round the result to four decimal places.

Evaluate the integral. ∫1∞dxxln⁡x\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }∫1∞​xlnxdx​

Evaluate the following integral. ∫013ln⁡2xxdx\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x∫01​x​3ln2x​dx

A particle moves on a straight line with velocity function v(t)=sin⁡atcos⁡3atv ( t ) = \sin a t \cos ^ { 3 } a tv(t)=sinatcos3at . Find its position function s=f(t)s = f ( t )s=f(t) if f(0)=0f ( 0 ) = 0f(0)=0 .

Evaluate the integral. ∫04xx+36dx\int _ { 0 } ^ { 4 } \frac { x } { x + 36 } d x∫04​x+36x​dx

Evaluate the integral. ∫6sin⁡2x1+cos⁡4xdx\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x∫1+cos4x6sin2x​dx

Evaluate the integral. ∫cos⁡(ln⁡18t)tdt\int \frac { \cos ( \ln 18 t ) } { t } d t∫tcos(ln18t)​dt

Evaluate the integral. ∫019xsin⁡πxdx\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x∫01​9xsinπxdx

Evaluate the integral. ∫7dx(x2+2x+2)2\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }∫(x2+2x+2)27dx​

Evaluate the integral. ∫0π/2cos⁡6xdx\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x∫0π/2​cos6xdx

Find the average value of the function f(x)f ( x )f(x) in the interval [−π,π][ - \pi , \pi ][−π,π] . f(x)=sin⁡6xcos⁡5xf ( x ) = \sin ^ { 6 } x \cos ^ { 5 } xf(x)=sin6xcos5x

Evaluate the integral using the indicated trigonometric substitution. ∫dxx2x2−49;x=7sec⁡θ\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 49 } } ; x = 7 \sec \theta∫x2x2−49​dx​;x=7secθ

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. ∫01dx2x+4;n=7\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7∫01​2x+4dx​;n=7

Evaluate the integral. ∫4y3eydy\int 4 y ^ { 3 } e ^ { y } d y∫4y3eydy

Evaluate the integral or show that it is divergent. ∫1∞ln⁡xx6dx\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 6 } } d x∫1∞​x6lnx​dx

Evaluate the integral. ∫013x03dx\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x∫01​x033​dx

Use long division to evaluate the integral. ∫01x3+12x2−36x+1x2+4x−12dx\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x∫01​x2+4x−12x3+12x2−36x+1​dx The choices are rounded to 3 decimal places.

Evaluate the integral using an appropriate trigonometric substitution. ∫02x24−x2dx\int _ { 0 } ^ { \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { 4 - x ^ { 2 } } } d x∫02​​4−x2​x2​dx

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Use the Trapezoidal Rule to approximate $\int _ { 2 } ^ { 3 } e ^ { 3 / x } d x$ for $n = 4$ . Round the result to four decimal places.

Evaluate the integral. $\int _ { 1 } ^ { \infty } \frac { d x } { x \ln x }$

Evaluate the following integral. $\int _ { 0 } ^ { 1 } \frac { 3 \ln 2 x } { \sqrt { x } } d x$

A particle moves on a straight line with velocity function $v ( t ) = \sin a t \cos ^ { 3 } a t$ . Find its position function $s = f ( t )$ if $f ( 0 ) = 0$ .

Evaluate the integral. $\int _ { 0 } ^ { 4 } \frac { x } { x + 36 } d x$

Evaluate the integral. $\int \frac { 6 \sin 2 x } { 1 + \cos ^ { 4 } x } d x$

Evaluate the integral. $\int \frac { \cos ( \ln 18 t ) } { t } d t$

Evaluate the integral. $\int _ { 0 } ^ { 1 } 9 x \sin \pi x d x$

Evaluate the integral. $\int \frac { 7 d x } { \left( x ^ { 2 } + 2 x + 2 \right) ^ { 2 } }$

Evaluate the integral. $\int _ { 0 } ^ { \pi / 2 } \cos ^ { 6 } x d x$

Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$ . $f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { d x } { x ^ { 2 } \sqrt { x ^ { 2 } - 49 } } ; x = 7 \sec \theta$

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $\int _ { 0 } ^ { 1 } \frac { d x } { 2 x + 4 } ; \quad n = 7$

Evaluate the integral. $\int 4 y ^ { 3 } e ^ { y } d y$

Evaluate the integral or show that it is divergent. $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 6 } } d x$

Evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { 3 } { x ^ { 03 } } d x$

Use long division to evaluate the integral. $\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1 } { x ^ { 2 } + 4 x - 12 } d x$ The choices are rounded to 3 decimal places.

Evaluate the integral using an appropriate trigonometric substitution. $\int _ { 0 } ^ { \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { 4 - x ^ { 2 } } } d x$