Thread: Integration by substitution

∫√(1-x^2 ) dx

Letting x = sin θ,
dx/dθ = cos θ

∫√(1-x^2 ) dx
=∫√(1-sin^2⁡θ ) cos θ dθ
=∫cos^2 θ dθ
=∫ (1-sin^2⁡θ) dθ
=θ -2sinθ(-cosθ) + c
=θ +2sinθcosθ + c
=sin^(-1)⁡θ + 2x√(1-x^2 ) + c

Is this correct? Please help me to check whether it is correct or not, and point out my mistake, if any. Thank you.

Originally Posted by alexander9408

=∫√(1-sin^2⁡θ ) cos θ dθ
=∫cos^2 θ dθ

Hint: From here, do it by integration by parts twice.

-Dan

Hello, alexander9408!

Use the identity: .



. . . . .

. . . . .


Back-substitute: .