Methods of integration
Change the variable and the limits together, integrate by parts, split a fraction into partial fractions, pick a trigonometric substitution and decide whether an improper integral has a value.
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Guide
integration by substitution: write u for the inside part, replace dx using du, and for a definite integral change the limits to u-values too. Then you never need to go back to x.
integration by parts: ∫udv = uv − ∫vdu. Choose u as the part that gets simpler when differentiated (x), and dv as the part you can integrate (eˣ dx). partial fractions: a fraction with a factorised denominator is split into simple fractions, each easy to integrate.
For √(1 − x²) the substitution x = sin θ turns the root into cos θ. An improper integral has an infinite limit (or an infinite value of f): it is a limit, and it may converge to a number or diverge.