Question
Download Solution PDFWhat is \(\rm \int^\pi _0 ln\left(tan\frac{x}{2}\right) dx\) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula 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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