NIMCET 2012 — Mathematics PYQ

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Question

NIMCET 2012 — Mathematics PYQ

NIMCET | Mathematics | 2012

The value of integral $\int_{0}^{\pi/2} \log \tan x \, dx$ is:

Choose the correct answer:

Explanation

1. Let the integral be $I$ :

I = \int_{0}^{\pi/2} \log(\tan x) \, dx \quad \dots \text{(Equation 1)}

2. Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$ :

I = \int_{0}^{\pi/2} \log(\tan(\pi/2 - x)) \, dx

Since $\tan(\pi/2 - x) = \cot x$ :

I = \int_{0}^{\pi/2} \log(\cot x) \, dx \quad \dots \text{(Equation 2)}

3. Adding Equation 1 and Equation 2:

2I = \int_{0}^{\pi/2} [\log(\tan x) + \log(\cot x)] \, dx

4. Using the logarithmic property $\log A + \log B = \log(AB)$ :

2I = \int_{0}^{\pi/2} \log(\tan x \cdot \cot x) \, dx

Since $\tan x \cdot \cot x = 1$ :

2I = \int_{0}^{\pi/2} \log(1) \, dx

5. Final Calculation:

Because $\log(1) = 0$ :

2I = \int_{0}^{\pi/2} 0 \, dx

2I = 0

I = 0

Correct Option:

(d) 0

Choose the correct answer:

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