Integration by substitution, integration by parts, integration using partial fractions, and integrating expressions of the form f prime over f and products reducible by a trigonometric identity.

What this dot point is asking

OCR wants you to integrate harder expressions using substitution (including changing the limits for a definite integral), integration by parts, and partial fractions, to recognise the $\dfrac{f'(x)}{f(x)}$ pattern that integrates to a logarithm, and to use trigonometric identities to integrate squared functions such as $\sin^2 x$.

The answer

Integration by substitution

Substitution reverses the chain rule. Pick an inner expression as $u$, replace $dx$ using $du = \dfrac{du}{dx}\,dx$, rewrite the whole integral in $u$, integrate, and (for an indefinite integral) substitute back.

For a definite integral it is cleaner to change the limits to values of $u$ rather than substituting back.

The logarithm pattern

When the numerator is the derivative of the denominator, the integral is a logarithm. If the numerator is a constant multiple of the derivative, adjust by that constant. This pattern is worth spotting before reaching for a full substitution, because it gives the answer in one line.

Integration by parts

Integration by parts reverses the product rule. Choose $u$ to be the factor that simplifies when differentiated, and $\dfrac{dv}{dx}$ to be the factor you can integrate.

A useful guide for choosing $u$: logarithms and powers of $x$ simplify on differentiating, so they make good choices for $u$; exponentials and sines stay manageable when integrated, so they make good choices for $\dfrac{dv}{dx}$.

Partial fractions

A rational function with a factorised denominator is integrated by splitting it into partial fractions first, after which each piece becomes a logarithm.

Choosing the right technique

With several techniques available, a quick decision tree helps. If you spot $\dfrac{f'(x)}{f(x)}$, write down the logarithm immediately. If the integrand is a product where one factor is the derivative of the inside of the other, use substitution. If it is a product of two unrelated functions (such as $x$ times an exponential or a trigonometric function), use parts. If it is a rational function, split into partial fractions. If it contains $\sin^2 x$ or $\cos^2 x$, apply a double angle identity first. Recognising the form quickly is worth as much as the technique itself in a timed paper.

Examples in context

A trigonometric identity first

You cannot integrate $\sin^2 x$ directly, but the double angle identity turns it into something you can.

Integrating with partial fractions

Try this

Q1. Find $\displaystyle\int \dfrac{6x^2}{x^3 + 2}\,dx$. [2 marks]

• Cue. Numerator is twice the derivative of the denominator: $2\ln|x^3 + 2| + c$.

Q2. Find $\displaystyle\int x\sin x\,dx$ by parts. [3 marks]

• Cue. $u = x$, $\frac{dv}{dx} = \sin x$, so $-x\cos x + \int\cos x\,dx = -x\cos x + \sin x + c$.