Methods of integration

The essentials

Many integrands are not in the standard table and need a method.

Substitution is used when the integrand contains a function and (a multiple of) its derivative. Put t = g(x), so dt = g′(x) dx, and rewrite the whole integral in t. Example: in ∫ 2x cos(x²) dx, put t = x² to get ∫ cos t dt = sin(x²) + C.

Partial fractions split a proper rational function into simpler fractions. Example: 1/[x(x + 1)] = 1/x − 1/(x + 1), whose integral is log|x| − log|x + 1| = log|x/(x + 1)| + C. If the numerator's degree is not less than the denominator's, divide first.

Integration by parts: ∫ u v dx = u ∫ v dx − ∫ [u′ ∫ v dx] dx. Choose u using the ILATE order (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential); the function earlier in this list is usually taken as u.

Key points

• ∫ x eˣ dx = (x − 1)eˣ + C.
• ∫ log x dx = x log x − x + C (take log x as u and 1 as v).
• ∫ eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.
• ∫ 1/(x² − a²) dx = (1/2a) log|(x − a)/(x + a)| + C.
• ∫ 1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + C.
• ∫ f′(x)/f(x) dx = log|f(x)| + C.

Exam tips

• Look for "derivative present" patterns first; substitution is fastest when it works.
• Always check the degree before using partial fractions.

Common mistakes

• Forgetting to change dx into dt during substitution.
• Choosing the wrong u in integration by parts, which makes the new integral harder.