StudySquare
Methods of integration for SQA Higher Maths
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.
1
Evaluate ∫1/(x + 5)dx.
∫1/(x + 5)dx = ln|x + 5| + c, where c is the constant of integration.
2
Solve the following integral: ∫5x⁴/(x⁵ + 10)dx.
∫5x⁴/(x⁵ + 10)dx = ln|x⁵ + 10| + c, where c is the constant of integration.
3
Determine the following integral: ∫(3x + 10)/(2x + 1)²dx.
Rewriting the integral in partial fractions gives ∫(3x + 10)/(2x + 1)²dx = ∫32/2(2x + 1) + 17/2(2x + 1)²dx. Integrating this gives (3/4)(ln|2x + 1| + 1/(2x + 1)) − 5/(2x + 1) + c, where c is the constant of integration.
4
Solve the following integral: ∫(7x − 20)/(20x + 1)dx.
Rewriting the integral in partial fractions gives ∫(7x − 20)/(20x + 1)dx = ∫7/20 − 407/20(20x + 1)dx. Integrating this gives 7x/20 − 407ln|20x + 1|/400 + c, where c is the constant of integration.
5
Determine the following integral ∫(30m² + 24m³)/(10m³ + 6m⁴)dm.
∫(30m² + 24m³)/(10m³ + 6m⁴)dm = ln|10m³ + 6m⁴| + c, where c is the constant of integration.
ln|10m³ + 6m⁴| + c
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