Power Formulas

Joey F. Valdriz

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A. Power of a Variable

The integral of a variable to a power is the variable to a power increased by one and divided by the new power. Formula:

Examples

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EXAMPLE: Evaluate SOLUTION:

EXAMPLE: Evaluate

SOLUTION:

B. Constants

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A constant may b e written either before or after the integral sign.

Formula:

Examples

5

Example: Evaluate Solution:

Example: Evaluate Solution:

C. Sums

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The integral of an algebraic sum of differentiable functions is the same as the algebraic sum of the integrals of these functions taken separately; that is, the integral of a sum is the sum of the integrals.

Formula:

Examples

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Example: Evaluate Solution:

Example: Evaluate Solution:

D. Power of a Function

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The integral of a function raised to a power is found by the following steps: 1. Increase the power of the function by 1. 2. Divide the result of step 1 by this increased power. 3. Add the constant of integration.

Formula:

Examples

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Example: Evaluate

Solution: Let so that or Then by substitution,

Therefore,

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Example: Evaluate

Solution: Let so that We find dx in the integral but not 3 dx. A 3 must be included in the integral to fulfill

the requirements of du. In words, this means the integral needs du so

that the formula may be used. Therefore, we write and recalling that a constant may be carried across the integral sign, we write

Notice that we needed 3 in the integral for du, and we included 3 in the integral; we then compensated for the 3 by multiplying the integral by 1/3. Then

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Example: Evaluate

Solution: Let so that Then

E. Quotient (Method 1: Putting the quotient into the form of the power of a

function)

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Examples

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Example: Evaluate Solution:

Let so that The factor 2 is used in the integral to give du and is compensated for by multiplying the integral by 1/2. Therefore,

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E. Quotient (Method 2: Operations with logarithms)

In the previous formulas for integration of a function, the exponent was not allowed to

be -1. In the special case of where

we would have applied the following formula:

Formula:

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E. Quotient (Method 3: A special case in which the quotient must be

simplified to use the sum rule)

In the third method for solving integrals of quotients, we find that to integrate an algebraic function with a numerator that is not of lower degree than its denominator, we proceed as follows: Change the integrand into a polynomial plus a fraction by dividing the denominator into the numerator. After this is accomplished, apply the rules available.

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Example: Evaluate Solution: Divide the denominator into the numerator so that

Integrating each term separately, we have

and and

Then, by substitution, we find that

Where

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Example: Evaluate Solution: The numerator in not of lower degree than the denominator; therefore, we divide and find that

Integrating each term separately, we find that and

Therefore,

where

Recall

Previous learning:

If f(x) = ex then f’(x) = ex

If f(x) = ax then f’(x) = (ln a)ax

If f(x) = ekx then f’(x) = kekx

If f(x) = akx then f’(x) = k(ln a)akx

This leads to the following formulas:

Indefinite Integrals of

Exponential Functions

19 ln

k xkx a

a dx Ck a

ln

xx a

a dx Ca

kxkx e

e dx Ck

x xe dx e C

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Shortcuts: Integrals of

Expressions Involving ax + b

1

1( 1)

nn ax b

ax b dx C na n

1 1

lnax b dx ax b Ca

1ax b ax be dx e Ca

1

ln

ax b ax bc dx c Ca c

Examples

9 99 t t te dt e dt e C 9

9

9

tt e

e dt C

55 443 3

5

4

uu e

e du C

55

444

35

12

5

u ue e CC

You Do

52 xdx

52

5(ln2)

x

C

Indefinite Integral of x-1

The rule is

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1 1lnx dx dx x C

x

Note: if x takes on a negative value, then ln x will be undefined. The absolute value sign keeps that from happening.

Example

4l1

n4

4dx dxx

xx

C

You Do:

25 xe dxx

21

5ln2

xx e C

Integrating Exponential Functions.mp4

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Exercises

1. 𝑥𝑒−𝑥2𝑑𝑥

2. 3𝑒4𝑥𝑑𝑥

3. 𝑒𝑥44𝑥3𝑑𝑥

4. 4𝑑𝑥

sec 𝑥𝑒𝑠𝑖𝑛𝑥

5. 72𝑥+3

Evaluate the following.

1. 𝑒1

2𝑥𝑑𝑥

2. 6𝑒−3𝑥𝑑𝑥

3. 𝑒3𝑥+1𝑑𝑥

4. 5𝑥𝑒−𝑥2𝑑𝑥

5. 𝑒1𝑥

𝑥2𝑑𝑥

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assignment

Evaluate the following functions.

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1.

2.

3.

4.

5.

3

3 2

t

t

e dt

e

Evaluate the following functions.

ADDITIONAL Exercises

6.

7.

8.

9. 𝑥+1

𝑥𝑑𝑥

10. 28

dxx

x

2

1

3(ln )xdx

x

Thank You!

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