Quick questions on Integration techniques: substitution, by parts and partial fractions - OCR A-Level Maths A

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.

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

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.

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

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

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