Maharashtra State BoardHSC Arts 12th Board Exam
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Evaluate : ∫0π41+sin2x⋅dx - Mathematics and Statistics

Sum

Evaluate : `int_0^(pi/4) sqrt(1 + sin 2x)*dx`

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Solution

`int_0^(pi/4) sqrt(1 + sin 2x)*dx`

= `int_0^(pi/4) sqrt(sin^2x + cos^2x + 2 sin x cos x)*dx`

= `int_0^(pi/4) sqrt((sinx + cosx)^2)*dx`

= `int_0^(pi/4) (sinx + cosx)*dx`

= `int_0^(pi/4) sinx*dx + int_0^(pi/4) cosx*dx`

= `[ - cos x]_0^(pi/4) + [sin x]_0^(pi/4)`

= `[- cos  pi/4 - (- cos 0)] + [sin  pi/4 - sin 0]`

= `-(1)/sqrt(2) + 1 + (1)/sqrt(2) - 0`
= 1.

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