Solving Integrals by Trigonometric substitution/Recognising cases

The most important thing about using a technique to integrate is recognising when it will be most useful.

Examples

Below are examples of possible substitutions.

• $x=a\sin u$
  • $dx=a\cos udu$
  • $\sqrt{a^2-x^2}=a\cos u$
  • Identity: $a^2{\sin }^2\theta +a^2{\cos }^2\theta =a^2$

• $x=a\cos u$
  • $dx=-a\sin udu$
  • $\sqrt{a^2-x^2}=a\sin u$
  • Identity:: $a^2{\sin }^2\theta +a^2{\cos }^2\theta =a^2$

• $x=a\tan u$
  • $dx=a{\sec }^{2}udu$
  • $\sqrt{a^2+x^2}=a\sec u$
  • Identity:: $a^2{\tan }^2\theta +a^2=a^2{\sec }^2\theta$

• $x=a\sec u$
  • $dx=a\tan u\sec udu$
  • $\sqrt{x^2-a^2}=a\tan u$
  • Identity:: $a^2{\tan }^2\theta +a^2=a^2{\sec }^2\theta$