Question

Evaluate the following integrals using properties of integration:

`int_0^(sin^2x) sin^-1 sqrt("t")  "dt" + int_0^(cos^2x) cos^-1 sqrt("t")  "dt"`

Answer 1

I₁ = `int_0^(sin^2x) sin^-1 sqrt("t")  "dt"`

Put `si^-1 sqrt("'t") = theta`

`sqrt("t")` = sin θ

t	0	sin2x
θ	0	x

`1/(2sqrt("t")) "dt"` = cos θ dθ

dt = `2sqrt("t") cos theta  "d"theta`

= 2 sin θ cos θ dθ

dt = sin 2θ dθ

I₁ = `int_0^x theta  sin2theta  "d"theta`

= `[ (- thetacos2theta)/2 + (sin2theta)/4]_0^x`

= `(-x cos 2x)/2 + (sin2x)/4`  ......(1)

I₁ =  `int_0^(cos^2x)  cos^-1 sqrt("t") "dt"`

Put `cos^-1 sqrt("t")` = θ

`sqrt("t"")` = cos θ

`1/(2sqrt("t"))  "dt"` = – sin θ dθ

dt`- 2sqrt("t")  sin theta  "d"theta`

= – 2cos θ sin θ dθ
dt = – sin 2θ dθ

I₂ = `int_0^(cos^x) cos^-1  sqrt("t")  "dt"`

= `int_(pi/2)^x - theta sin 2theta  "d"theta`

t	0	cos2x
θ	`pi/2`	x

= `- [(- theta cos 2theta)/2+ (sin theta)/4]_(pi/2)^x`

= `[(theta cos2theta)/2 - (sin 2theta)/4]_(p/2)^x`

= `(x cos 2x)/2 - (sin 2x)/4 + pi/4`   ........(2)

I = I₁ + I₂

= `(- x cos 2x)/2 + (sin 2x)/4 + (x cos2x)/2 - (sin 2x)/4 + pi/4`

I = `pi/4`