\(\rm \int^\pi _0 ln\left(tan\frac{x}{2}\right)dx\) किसके बराबर है?

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Question

This question was previously asked in

NDA 02/2021: Maths Previous Year paper (Held On 14 Nov 2021)

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Answer 1

प्रयुक्त सूत्र:

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

tan(π - θ) = - tan θ

गणना:

माना

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

प्रयुक्त सूत्र के अनुसार

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

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

⇒ I = -I

⇒ 2I = 0

⇒ I = 0

∴समाकल\(\int^π _0 ln\left(tan\frac{x}{2}\right)dx\) का मान 0 के बराबर है