Integration by substitution and by parts · Year 13

In Year 13 you learn: integration by substitution.

When a function is not a simple power, you need a technique. You might spot a reverse chain rule, substitute a new variable, integrate by parts or use partial fractions. Choosing the right technique is half the skill.

Key facts

• Reverse chain rule: ∫ f′(x)(f(x))^n dx = ((f(x))^(n+1))/(n + 1) + c and ∫ (f′(x))/(f(x)) dx = ln|f(x)| + c.
• Standard results: ∫ e^(kx) dx = (e^(kx))/k, ∫ cos(kx) dx = (sin(kx))/k, ∫ sin(kx) dx = −(cos(kx))/k, ∫ 1/(ax + b) dx = 1/a ln|ax + b|.
• Substitution: choose u and replace dx using du/dx. Change the limits to u-values too.
• Partial fractions: split (px + q)/((x + a)(x + b)) into A/(x + a) + B/(x + b) before integrating.
• Powers of trig: use sin^2 x = (1 − cos 2x)/2 and cos^2 x = (1 + cos 2x)/2.

Worked examples