H: Integration: Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively. (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae).

The idea

Integration by substitution reverses the chain rule. For ∫ f(g(x))g'(x) dx, let u = g(x), then du = g'(x) dx, giving ∫ f(u) du. For definite integrals, change limits. Example: ∫ 2x cos(x²) dx, let u = x², du = 2x dx, giving ∫ cos u du = sin u + C = sin(x²) + C. Integration by parts reverses the product rule: ∫ u dv/dx dx = uv − ∫ v du/dx dx. Choose u and dv wisely (e.g., u = x, dv = e^x dx). Example: ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C. Integration by parts may need more than one application, e.g. ∫ x² e^x dx requires two applications; reduction formulae are excluded. Some integrals reappear after applying by parts and are solved algebraically.

What earns the marks

• Selects an appropriate substitution u = g(x) such that the integral transforms into a standard form, and correctly computes du = g'(x) dx.
• Changes the limits of integration when evaluating a definite integral by substitution, or substitutes back to the original variable for indefinite integrals.
• Applies integration by parts using the formula ∫ u dv = uv − ∫ v du, choosing u and dv to simplify the resulting integral.
• Performs integration by parts more than once when necessary, for example for integrals involving x² e^x or x² sin x, until the integral is manageable.
• Recognises that integration by substitution is the inverse of the chain rule and integration by parts is the inverse of the product rule, and explains this relationship.

Common mistakes and how to put them right

• Forgetting to change the limits when using substitution in a definite integral.

  Put it right: Either convert the limits to u-values or substitute back before evaluating.

• Choosing u and dv poorly in integration by parts, leading to a more complicated integral.

  Put it right: Use the guideline of choosing u as a function that simplifies when differentiated (e.g. ln x, x, x²) and dv as something easily integrated (e.g. e^x, sin x).

• Omitting the constant of integration for indefinite integrals.

  Put it right: Add + C for indefinite integrals.

• Attempting to use substitution when the derivative of the inner function is not present (up to a constant factor).

  Put it right: Check that the integrand contains g'(x) or a multiple of it; otherwise, the substitution may not work.

Exam tips

• When using substitution, clearly state u and du, and show the transformed integral before integrating.
• For integration by parts, write down u, dv, du and v explicitly, then apply the formula step by step.
• If a question asks for 'more than one application', be prepared to apply integration by parts repeatedly, simplifying each time, and watch for the integral reappearing (then solve algebraically).