Question

Prove that:

$\displaystyle\int^{\pi/2}_0 \dfrac{\sqrt{ \cos\theta}}{1+ \cos^2\theta} d\theta = \dfrac{\pi}{4}$
​

Answer 1

Answer:

Correct option is

A

I=4

Consider the given integral.

I=∫

0

π

2+2cosθ

dθ

We know that

cosx=2cos

2

2

x

−1

Therefore,

I=∫

0

π

2+2(2cos

2

2

θ

−1)

dθ

I=∫

0

π

2+4cos

2

2

θ

−2

dθ

I=∫

0

π

4cos

2

2

θ

dθ

I=2∫

0

π

cos

2

θ

dθ

I=2

⎝

⎜

⎜

⎛

2

1

sin

2

θ

⎠

⎟

⎟

⎞

0

π

I=4(sin

2

θ

)

0

π

I=4(sin

2

π

−sin0)

I=4(1−0)

I=4

Hence, this is the answer.