Methods of integration for Edexcel A-level Maths

Methods of integration

This page covers the following topics:

1. Integration by parts
2. Integration by substitution
3. Integrating partial fractions

Integration by parts is a special method of integration which is used when two functions are being multiplied. They are defined as u and v accordingly, and the given formula is used to evaluate the interval.

Integration by parts

Integration by substitution is a method to evaluate integrals by changing the variables. This method can be used when the integral is of the given form.

Integration by substitution

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 ∫1/(x + 5)dx.

∫1/(x + 5)dx = ln|x + 5| + c, where c is the constant of integration.

Evaluate ∫1/(x + 5)dx.

2

Use integration by parts to find ∫(5 + x)cos(x)dx.

Let u = 5 + x and dv = cos(x)dx.
Then, du = dx and v = sin(x).
∫ (5 + x)cos(x)dx = (5 + x)sin(x) − ∫sin(x)dx =
= (5 + x)sin(x) + cos(x) + c, where c is the constant of integration

(5 + x)sin(x) + cos(x) + c

Use integration by parts to find ∫(5 + x)cos(x)dx.

3

Determine the following integral: ∫3x²sin(x³ − 10x) − 10sin(x³ − 10x)dx.

The integral can be rewritten as ∫(3x² − 10)sin(x³ − 10x)dx. To integrate by substitution, let u = x³ − 10x and du = (3x² − 10)dx. So, ∫(3x² − 10)sin(x³ − 10x)dx = ∫sin(u)du = −cos(u) + c = −cos(x³ − 10x) + c, where c is the constant of integration.

Determine the following integral: ∫3x²sin(x³ − 10x) − 10sin(x³ − 10x)dx.

4

Evaluate ∫xeˣ.

Let u = x and dv = eˣdx. Then, du = 1dx and v = eˣ. So, ∫xeˣ = xeˣ − ∫eˣdx = xeˣ − eˣ + c, where c is the constant of integration.

Evaluate ∫xeˣ.

5

Integrate the area of the given rectangle.

The integral is ∫15x²sin(8 − 5x³)dx. To integrate by substitution, let u = 8 − 5x³ and du = −15x²du. So, ∫15x²sin(8 − 5x³)dx = ∫−sin(u)dx = cos(u) + c = cos(8 − 5x³) + c, where c is the constant of integration.

Integrate the area of the given rectangle.

End of page