Integration Techniques — Last Minute Summary

Reduction Formulae — Target Pattern

$I_n = \int f(x,n)\,dx$

Integration by parts → $I_n = A(n)I_{n-k} + B(x)$

Common Recurrences

$I_n$	Recurrence
$\int_0^{\pi/2} \sin^n x\,dx$	$n I_n = (n-1)I_{n-2}$
$\int_0^{\pi/2} \cos^n x\,dx$	$n I_n = (n-1)I_{n-2}$
$\int_0^1 x^n e^x\,dx$	$I_n = e - nI_{n-1}$
$\int_0^1 (1-x^2)^{n/2}\,dx$	$(n+2)I_n = (n+1)I_{n-2}$

Boundary Values

$I_0 = \int_a^b 1\,dx = b-a$

$I_1: \text{ compute directly}$

$\int_0^{\pi/2} \sin^n x\,dx: I_0 = \frac{\pi}{2},\; I_1 = 1$

Integration by Parts (LIATE)

$u = \text{Log} \to \text{Inverse trig} \to \text{Algebraic} \to \text{Trig} \to \text{Exponential}$

Partial Fractions Quick Reference

Denominator Factor	Term Form
$x-a$	$\frac{A}{x-a}$
$(x-a)^2$	$\frac{A}{x-a} + \frac{B}{(x-a)^2}$
$x^2 + a^2$	$\frac{Ax+B}{x^2+a^2}$

Standard Integrals

$\int \frac{1}{x}\,dx = \ln|x|$

$\int \tan x\,dx = -\ln|\cos x|$

$\int \frac{1}{x^2 + a^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a}$

$\int \frac{x}{x^2 + a^2}\,dx = \frac{1}{2}\ln(x^2 + a^2)$

Trap Checklist

❌ Forgetting $+C$ for indefinite integrals

❌ Substitution: forgetting to change limits

❌ Partial fractions: need $\frac{Bx+C}{x^2+a^2}$, not just $\frac{B}{x^2+a^2}$

❌ Degree of numerator $\ge$ denominator → divide first

❌ Reduction: always state $I_0$ or $I_1$ before iterating