Quick questions on Integration techniques explained: H2 Mathematics Calculus

The reverse of the standard derivatives, including:

Substitution reverses the chain rule. Choose $u$ so that $\dfrac{du}{dx}$ appears (up to a constant) in the integrand, rewrite the whole integral in $u$ (including $dx$), integrate, then revert to $x$. For definite integrals, change the limits to $u$-values.

A proper rational function with a factorable denominator splits into simpler fractions, each integrable as a logarithm or power. For example $\dfrac{1}{(x-1)(x+2)}$ becomes $\dfrac{A}{x-1} + \dfrac{B}{x+2}$, and each term integrates to a logarithm.

When substituting, replace $dx$ in terms of $du$; integrating with a leftover $dx$ is wrong.

Choose $u$ to simplify on differentiation (logs, powers); a bad choice makes the new integral harder.

If the numerator degree is at least the denominator degree, do polynomial division first.

Find $\displaystyle\int (2x + 1)^4\,dx$. [2 marks]

Find $\displaystyle\int x e^x\,dx$. [3 marks]

Express $\dfrac{1}{x(x+1)}$ in partial fractions. [2 marks]