Trigonometric Substitution

Unit 7.4 of Math 152

Motivating problem

• $\int \sqrt{a^2-x^2}dx$

  • We can’t integrate this :(

    • but if we make the integral in terms of angle because this is a semi circle, we can use the pythagorean identity to rewrite it as
      • $x=a\sin \theta ,⟹1-{\sin }^2\theta ={\cos }^2\theta$
        • this allows us to get rid of the root sign with algebra and replace it with absolute value.
  • let : $x=\sin \theta$
    • $dx=\cos \theta d\theta$
      • $I=\int \sqrt{1-{\sin }^2\theta }\cos \theta d\theta$
        • => using the pythagorean identity
          • => $\int \sqrt{{\cos }^2\theta }\cos \theta d\theta$
            • => $\int {\cos }^2\theta d\theta$
              • => using the reduction formula
                • => $\frac{1}{2}\sin \theta \cos \theta +\frac{1}{2}\theta +C$

                • writing in terms of x

                  • =>