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Integration

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Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 3

Integration

The big idea: Integration in Paper 3 is pattern recognition: spot which form the integrand is in, rewrite it if needed (identity, partial fractions, substitution, parts), and then use a standard result.

Summary — What This Chapter Covers

Standard integrals: e^{ax + b}, 1/(ax + b), sin(ax + b), cos(ax + b), sec²(ax + b) and 1/(a² + x²).
Use double-angle formulae to integrate sin²x and cos²x, and sec²x − 1 for tan²x.
Split rational functions into partial fractions; recognise kf′(x)/f(x), whose integral is k ln|f(x)|.
Use a given substitution, changing the limits as well for a definite integral.
Integration by parts: ∫u (dv/dx) dx = uv − ∫v (du/dx) dx.

1. Standard Integrals and Trigonometric Identities

Standard forms

Reverse differentiation, extended

Each standard integral reverses a standard derivative; with a linear inner function ax + b, divide by a. Squared trig functions are integrated by first using an identity to remove the square.

Standard integrals ∫e^{ax + b} dx = (1/a)e^{ax + b} + c    ∫1/(ax + b) dx = (1/a) ln|ax + b| + c
∫sin(ax + b) dx = −(1/a)cos(ax + b) + c    ∫cos(ax + b) dx = (1/a)sin(ax + b) + c
∫sec²(ax + b) dx = (1/a)tan(ax + b) + c    ∫1/(a² + x²) dx = (1/a)tan⁻¹(x/a) + c

Identities sin²x = ½(1 − cos 2x) cos²x = ½(1 + cos 2x) tan²x = sec²x − 1

Worked Example 1

Find ∫1/(4 + 9x²) dx.

Worked Example 2

Find the exact value of ∫ from 0 to π/4 of cos²(2x) dx.

Practice Question

Find ∫sin²x dx.

2. Integration Using Partial Fractions

Partial fractions

Split first, then each piece is a log

Use the partial fraction forms from Algebra. Each A/(ax + b) integrates to (A/a) ln|ax + b|; a repeated term C/(ax + b)² integrates to −C/(a(ax + b)). Also recognise integrands already of the form kf′(x)/f(x).

Log forms ∫f′(x)/f(x) dx = ln|f(x)| + c e.g. ∫x/(1 + x²) dx = ½ ln(1 + x²) + c, ∫tan x dx = −ln|cos x| + c = ln|sec x| + c

Worked Example

Show that ∫ from 2 to 3 of (x + 5)/((x − 1)(x + 1)²) dx = (3/2) ln(3/2) − 1/6.

Practice Question

Find ∫ x/(1 + x²) dx and ∫ tan x dx.

3. Integration by Substitution

Substitution

Changing the variable

The substitution is always given in this paper. Write dx in terms of du, replace every x, and for a definite integral change the limits to u-values so you never have to substitute back.

Method u = g(x) ⇒ du = g′(x) dx; new limits u = g(a), u = g(b)

Worked Example

Use the substitution u = x + 1 to find the exact value of ∫ from 0 to 3 of x√(x + 1) dx.

Practice Question

Use the substitution u = sin x to find ∫sin²(2x) cos x dx.

4. Integration by Parts

Parts

Integrating a product

⁻¹ ² ^−x

Formula ∫u (dv/dx) dx = uv − ∫v (du/dx) dx

Worked Example 1

Find ∫x sin 2x dx.

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Integration

Integration

Reverse differentiation, extended

Split first, then each piece is a log

Changing the variable

Integrating a product

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