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Methods of integration for SQA Higher Maths

Methods of integration

This page covers the following topics:

1. Integrating partial fractions

When the integrand is in the form of a proper fraction, the integral should be rewritten as partial fractions and then integrated using the given result.

Integrating partial fractions

1

Evaluate โˆซ1/(x + 5)dx.

โˆซ1/(x + 5)dx = ln|x + 5| + c, where c is the constant of integration.

Evaluate โˆซ1/(x + 5)dx.

2

Solve the following integral: โˆซ5xโด/(xโต + 10)dx.

โˆซ5xโด/(xโต + 10)dx = ln|xโต + 10| + c, where c is the constant of integration.

Solve the following integral: โˆซ5xโด/(xโต + 10)dx.

3

Determine the following integral: โˆซ(3x + 10)/(2x + 1)ยฒdx.

Rewriting the integral in partial fractions gives โˆซ(3x + 10)/(2x + 1)ยฒdx = โˆซ32/2(2x + 1) + 17/2(2x + 1)ยฒdx. Integrating this gives (3/4)(ln|2x + 1| + 1/(2x + 1)) โˆ’ 5/(2x + 1) + c, where c is the constant of integration.

Determine the following integral: โˆซ(3x + 10)/(2x + 1)ยฒdx.

4

Solve the following integral: โˆซ(7x โˆ’ 20)/(20x + 1)dx.

Rewriting the integral in partial fractions gives โˆซ(7x โˆ’ 20)/(20x + 1)dx = โˆซ7/20 โˆ’ 407/20(20x + 1)dx. Integrating this gives 7x/20 โˆ’ 407ln|20x + 1|/400 + c, where c is the constant of integration.

Solve the following integral: โˆซ(7x โˆ’ 20)/(20x + 1)dx.

5

Determine the following integral โˆซ(30mยฒ + 24mยณ)/(10mยณ + 6mโด)dm.

โˆซ(30mยฒ + 24mยณ)/(10mยณ + 6mโด)dm = ln|10mยณ + 6mโด| + c, where c is the constant of integration.

ln|10mยณ + 6mโด| + c

Determine the following integral โˆซ(30mยฒ + 24mยณ)/(10mยณ + 6mโด)dm.

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