Evaluate the following integral using properties of integration. ∫sin^-1 √t dt for t ∈ [0,sin^2 x] + ∫cos^-1 √t dt for t ∈ [0,cos^2x]

Question

Evaluate the following integral using properties of integration.

\(\int\limits_{0}^{sin^2 x} sin^{-1} √t dt + \int\limits_{0}^{cos ^2 x} cos ^{-1 } √t dt \)

Answer 1

From the integral

Let I = \(\int\limits_{0}^{sin^2 x} sin^{-1} √t dt + \int\limits_{0}^{cos ^2 x} cos ^{-1 } √t dt \) ... (1)

Consider:

I₁ = \(\int\limits_{0}^{sin^2 x} sin^{-1} √t dt \)

Put t = sin² x 

Differentiate with respect to x 

dt = 2 sin x cos x dx 

dt = sin 2x dx 

t	0	sin2 x
x	0	x

Applying Bernoulli's formula