 # Methods of integration for SQA Higher Maths 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 End of page