Substitution and trig identities

Substitution is the chain rule backwards. Look for a function together with its derivative in the integrand. Put that function = t; its derivative together with dx becomes dt.

Example: ∫x² cos(x³) dx. The inside is x³, and its derivative 3x² is (nearly) there. Put t = x³, so dt = 3x² dx and x² dx = dt/3. The integral becomes (1/3)∫cos t dt = (1/3) sin(x³) + C.

Example: ∫(log x)/x dx. Put t = log x, dt = (1/x) dx. It becomes ∫t dt = t²/2 = (log x)²/2 + C.

Always write the final answer back in x.

A very useful pattern: ∫f′(x)/f(x) dx = log|f(x)| + C. If the top is the derivative of the bottom, the answer is the log of the bottom.

• ∫tan x dx = ∫sin x/cos x dx = –log|cos x| = log|sec x| + C.
• ∫cot x dx = log|sin x| + C.
• ∫sec x dx = log|sec x + tan x| + C, and ∫cosec x dx = log|cosec x – cot x| + C.

Trig identities turn powers and products into things you can integrate:

• sin²x = (1 – cos 2x)/2 and cos²x = (1 + cos 2x)/2.
• sin³x = (3 sin x – sin 3x)/4.
• Products to sums: 2 sin A cos B = sin(A + B) + sin(A – B), and similar.

For a linear inside, ∫f(ax + b) dx = (1/a) F(ax + b) + C: divide by the coefficient of x. So ∫e^(5x) dx = e^(5x)/5 + C.