Quick questions on Integration techniques: substitution and partial fractions (QCE Specialist Mathematics Unit 4)

Substitution reverses the chain rule. If the integrand contains a function and (a multiple of) its derivative, let $u$ be the inner function. With $u = g(x)$ and $du = g'(x)\,dx$,

A proper rational function with a factorisable denominator can be split into simpler fractions. For distinct linear factors,

Rotating the region under $y = f(x)$ between $x = a$ and $x = b$ about the $x$-axis produces a solid of volume

A good substitution makes the derivative of the inner function appear (up to a constant) elsewhere in the integrand. Telltale signs are a composite function with its inner derivative present, such as $6x$ alongside $3x^2 + 1$, or an expression of the form $\dfrac{g'(x)}{g(x)}$, whose integral is $\ln|g(x)|$. When the integrand is a single awkward function with no obvious inner derivative, a trigonometric substitution (for example $x = a\sin\theta$ to handle $\sqrt{a^2 - x^2}$) may be the intended route.

Fluent integration depends on recognising a small library of standard forms: $\displaystyle\int \dfrac{1}{x}\,dx = \ln|x| + C$, $\displaystyle\int \dfrac{1}{a^2 + x^2}\,dx = \dfrac{1}{a}\arctan\dfrac{x}{a} + C$, $\displaystyle\int \dfrac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\dfrac{x}{a} + C$, and $\displaystyle\int e^{kx}\,dx = \dfrac{1}{k}e^{kx} + C$. Many integration questions reduce, after a substitution or a partial-fraction split, to one of these forms, so the strategy is always to manipulate the integrand until it matches a standard result.