What is \(\rm \int^\pi _0 ln\left(tan\frac{x}{2}\right) dx\) equal to?

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NDA 02/2021: Maths Previous Year paper (Held On 14 Nov 2021)
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  1. 0
  2. \(\rm \frac{1}{2}\)
  3. 1
  4. 2

Answer (Detailed Solution Below)

Option 1 : 0
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Detailed Solution

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Formula used:

\(\rm \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx\)

tan(π - θ) = - tan θ  

Calculation:

Let

I = \(\int^π _0 ln\left(tan\frac{x}{2}\right)dx\)        ---(1)

According to the formula used

I = \(\int^π _0 ln\left(tan (π -\frac{x}{2})\right)dx\)

⇒ I = - \(\int^π _0 ln\left(tan\frac{x}{2}\right)dx\)

From equation (1)

⇒ I = -I

⇒ 2I = 0

⇒ I = 0

∴ The value of the integral \(\int^π _0 ln\left(tan\frac{x}{2}\right)dx\) is 0.

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