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Calc 8.2

The document provides instructions for evaluating several integrals using integration techniques like integration by parts, substitution, and recognizing patterns. The integrals involve trigonometric, exponential, logarithmic and other functions. The instructions tell the reader to identify the parts of the integral, find the antiderivative using Wolfram as needed, and then use a calculator to evaluate the final integral.

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K Cor
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75 views46 pages

Calc 8.2

The document provides instructions for evaluating several integrals using integration techniques like integration by parts, substitution, and recognizing patterns. The integrals involve trigonometric, exponential, logarithmic and other functions. The instructions tell the reader to identify the parts of the integral, find the antiderivative using Wolfram as needed, and then use a calculator to evaluate the final integral.

Uploaded by

K Cor
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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On which derivative rule is integration

What is the best choice

Evaluate the following integral.

Part 1: You have to understand the rule that:


Now, identify parts of the equation.

So, take the derivative of u for

u=7 x

du=7

u dv=uv v du
and

dv=e5 x

and the antiderivative of dv for

e5 x
5

( )

v=

You may want to use Wolfram for the antiderivative.


Then, youll have the equation as shown above.
Part 2: Just use the calculator to solve part 1. Be sure to use the subtraction sign
and the +C.

Evaluate the following integral.

Part 1: You have to understand the rule that:


Now, identify parts of the equation.
So, take the derivative of u for

u=12 t

u dv=uv v du
and

dv=et

du=12 and the antiderivative of dv for

v =e t .

You may want to use Wolfram for the antiderivative.


Then, youll have the equation as shown above.
Part 2: Just use the calculator to solve part 1. Be sure to use the subtraction sign
and the +C.

Evaluate the following integral.

Part 1: You have to understand the rule that:

Now, identify parts of the equation.


So, take the derivative of u for

v =4 cos

( 4x )

u=x

u dv=uv v du

and

dv=sin

( 14 ) x

du=1 and the antiderivative of dv for

You may want to use Wolfram for the antiderivative.


Then, youll have the equation as shown above.
Part 2: Use Wolfram. Calculator messes up..

Evaluate the following integral.

Part 1: You have to understand the rule that:

Now, identify parts of the equation.


So, take the derivative of u for

u=x

u dv=uv v du

and

dv=

1
x+7

du=1 and the antiderivative of dv for

.
You may want to use Wolfram for the antiderivative.
Then, youll have the equation as shown above.
Part 2: Use Wolfram. Calculator messes up..

v =2 x +7

Evaluate the integral.

Part 1: You have to understand the rule that:


Now, identify parts of the equation.
So, take the derivative of u for

u=4

du=4

u dv=uv v du
and

dv=sec 2

and the antiderivative of dv for

You may want to use Wolfram for the antiderivative.


Then, youll have the equation as shown above.
Part 2: Calculator.

v =tan .

Evaluate the following integral.

I cheated to answer this.


Part 1: The format is the same. The number in front of x^2 is simply half whatever
leads in the given equation. The ln is the same as the equation. Then the 5x comes
also from halving the lead.
Part 2: Calculator.

Evaluate the integral.

Part 1: Follow the same syntax as above. Subtract 1 from the exponent like shown
and plug in the pattern.
Part 2: Calculator.

Evaluate the following integral.

Part 1: You have to understand the rule that:

u dv=uv v du

Now, identify parts of the equation.


So, take the derivative of u for
for

u=4 xsin ( x ) and

du=4 (sin ( x )+ xcos ( x ))

dv=cos ( x )

and the antiderivative of dv

v =sin ( x ) .

You may want to use Wolfram for the antiderivative.


Now, use Wolfram to combine uv and v du before entering them into the equation
box.
Part 2: I used Wolram.

Evaluate the integral.

Part 1: You have to understand the rule that:


Now, identify parts of the equation.
So, take the derivative of u for

v=

u=t

du=2t

u dv=uv v du

and

3 t

dv=e

and the antiderivative of dv for

1 3 t
e
.
3

You may want to use Wolfram for the antiderivative.

Now, use Wolfram to combine uv and v du before entering them into the equation
box.
Part 2: I used Wolram.

Evaluate

Calculator.

Evaluate the following integral.

Part 1: You have to understand the rule that:


Now, identify parts of the equation.
So, take the derivative of u for

u=x 2

du=2 x

u dv=uv v du
and

dv=e11 x

and the antiderivative of dv for

v=

e11 x
11 .

You may want to use Wolfram for the antiderivative.


Now, use Wolfram to combine uv and v du before entering them into the equation
box.
Part 2: I used a calculator.

Evaluate the following definite integral.

Part 1: Break the equation up into


So,

du=7

and

u=7 x

and

cos ( x)

v =sin ( x )

On the first part of the rewrite combine uv and on the second combine du v
Part 2: Calculator.

Evaluate the definite integral.

Part 1:
So,

u=ln ( 8 x ) and

du=1/ x

and

dv=dx

v =x

On the left side combine uv and on the right side du v


Part 2: Calculator

Find the volume of the solid

Part 1: The limits are given. The inside is 2Pix f(x)


Part 2: u=2Pix and dv=sin(x) So, du=2PI and V=-cos(x)
On the left side combine uv and on the right du v
Part 3: Calculator
Use integration by parts

Use the reduction formulas

Calculator.
Use a substitution to reduce the following integral

Calculator.

Evaluate using the substitution

First two just copy.


Last one: calculator. You can enter the exponent within the parentheses such that
ln(x^10)
Prove that

Copy first parts.


Use Wolfram for derivative.
Calculator solves the next one. Shift and hyphen create the subscript.
The curves
If you get this check PDF #24 in folder 8.2
Evaluate the following integral.

I patched this together.


First part: u=tan^(-1)20x and dv=dx
Du=20/400x^(2)+1 and v=x
For the left side of part one combine uv, and the right side du v
Part 2: You just have to follow the pattern. Put an x out front and then follow it with
the function.
There is then a positive or negative sign. This depends on what you get for the right
side of the equation. If the combination results in a negative, then put a positive
(two negatives=positive). The fraction is putting one over twice the value of the
function number.
The last part is kind of a copy from the denominator in the right side. Just put an ln
in front of it.
Evaluate using a substitution followed

Ok Whatever is in front of the cos, multiply by 2 then divide that by the number in
the square root. Thats the number that goes out front. The wsinw sinw shit stays
the same.

Suppose a mass on a string

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