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Basic Integration Formulas Guide

The document provides formulas and examples for integrating several types of functions, including: 1) Basic integration formulas such as the integral of a variable u being u plus a constant C. 2) Integrals of logarithmic functions using the formula for integrating du/u. 3) Integrals of exponential functions using formulas for integrals of au and e^u. 4) Integrals of trigonometric functions by using trigonometric identities to reduce integrals to integrable forms. Practice problems with solutions are provided for each type of integral.

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Ronnanx Gomezx
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67 views3 pages

Basic Integration Formulas Guide

The document provides formulas and examples for integrating several types of functions, including: 1) Basic integration formulas such as the integral of a variable u being u plus a constant C. 2) Integrals of logarithmic functions using the formula for integrating du/u. 3) Integrals of exponential functions using formulas for integrals of au and e^u. 4) Integrals of trigonometric functions by using trigonometric identities to reduce integrals to integrable forms. Practice problems with solutions are provided for each type of integral.

Uploaded by

Ronnanx Gomezx
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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• The Basic Integration Formulas

1. ∫ du = u + C The integral of a differential variable u is u plus C

2. ∫ a du = a ∫ du The integral of a constant and a differential variable u

3. ∫ ( u +/- v ) du = ∫ u du +/- ∫ v du The integral of sum and difference

n+1
4. ∫ u du = u n+1 + C The power formula
n+1

PRACTICE PROBLEMS

1. Integrate ( 3x2 – 5 ) dx 2. Integrate ( x 8 – 4 x3 – x ) dx

3. Integrate ( x x1/2 - 5 )2 dx 4. Integrate ( x3 – 1 ) dx


x–1
5. Integrate ( x3 + 2x2 + x )1/2 dx

• The Integral of Logarithmic Functions

The formula ∫ du / u = ln u + C involves a numerator which is the derivative of the


denominator u represents any function involving any independent variable. The formula is
meaningless when u is negative, since the logarithms of negative numbers have not been
defined.

PRACTICE PROBLEMS:

1. Integrate ( 2x – 7 ) dx 2. Integrate ( 3 – 4x ) dx
4x2 – 6x + 20 x2 – 7x + 10

3. Integrate ( 4x2 – 8x – 9 ) dx 4. Integrate ( y2 +y ) dy


y–1 2x + 1

5. Integrate dy / y ( 1 + y2 ) 6. Integrate ( 1 – 3x )2 dx

7. Integrate cos t dt 8. Integrate ( x2 – 3 ) dx


3 + 3 sin t x3 – 9x + 5

9. Integrate ( x3 – 2x2 + 3 ) dx 10. Integrate dx / x ( x3 + 1 )


x–1

• The Integral of Exponential Function

Formulas:

1. ∫ a u du = ( a u / ln a ) + C provided a is greater than 0 and not equal to 1.

2. ∫ e u du = e u + C
PRACTICE PROBLEMS

1. Integrate ( 2 + e x ) dx 6. Integrate e – x /2 dx
ex

2. Integrate 3 x e x dx 7. Integrate e 2x dx
1 + ex

3. Integrate ( e x + e –x ) 2 dx 8. Integrate dy / 1 + ey

4. Integrate e x ( 1 + e x ) ½ dx 9. Integrate e 2x dx
1 + ex

5. Integrate ( 10 3x ) ½ dx 10. Integrate ( 2 – e –t ) 2 dt

• The Integral of Trigonometric Functions

The basic formulas for integration involving trigonometric functions are stated in terms of
appropriate pairs of functions. However, many trigonometric integrals can evaluated after the
transformations of the integrand, making use only of the familiar trigonometric formulas.
Trigonometric identities are employed to reduce the integrals to integrable forms.

PRACTICE PROBLEMS

1. Integrate ( 1 – cos x ) dx 6. Integrate cos3 x dx


sin 2 x 1 – sin x

2. Integrate ( cos 2 x ) dx 7. Integrate ( tan x – 1 )2 dx


1 + sin x

3. Integrate ( 1 – cos 2 x ) dx 8. Integrate ( sin3 y + cos3 y ) dy


1 + cos 2 x

4. Integrate ( sin x + 2 cos x )2 dx 9. Integrate ( sin3 x sin3 2x ) dx


sin x

5. Integrate ( sin2 x dx ) 10. Integrate ( sin5 y ) dy


1 – cos x cos2 y

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