Integration Methods
Cheat sheet.

Core rule

Watch for

Choose an easier remaining integral, retain the minus sign and boundary term, and solve algebraically when the original integral returns.

Core rule

Odd sine: save $\sin x\,dx$, use $u=\cos x$. Odd cosine: save $\cos x\,dx$, use $u=\sin x$. Both even: use half-angle identities.

Watch for

Track minus signs, frequency factors and intervals avoiding trigonometric poles. Tangent/secant patterns use $1+\tan^2x=\sec^2x$.

Core rule

Match $a^2-x^2$, $a^2+x^2$, or $x^2-a^2$ to sine, tangent, or a branch-aware secant substitution. Replace both the radical and differential.

Watch for

$\sqrt{u^2}=|u|$. Complete the square first when needed, state the angle branch, and transform bounds or return to $x$ before evaluation.

Core rule

Divide first if improper as a rational expression. Include every power of repeated factors; use linear numerators over irreducible quadratics.

Watch for

A rational expression with numerator degree too large is different from an improper integral. Logs need absolute values when their arguments may be negative on a domain interval.

Core rule

Watch for

Inverse hyperbolic functions are not reciprocals. Preserve the chosen real domain and inner scaling factors.