If , then the value of is equal…

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

Problem

If f(x)=−∫0xlog⁡(cos⁡t) dtf(x) = -\int_0^x \log (\cos t) \, dt , then the value of f(x)−2f(π4+x2)+2f(π4−x2)f(x) - 2f\left(\frac{\pi}{4} + \frac{x}{2}\right) + 2f\left(\frac{\pi}{4} - \frac{x}{2}\right) is equal to (1) −xlog⁡2-x \log 2 (2) x2log⁡2\frac{x}{2} \log 2 (3) x3log⁡2\frac{x}{3} \log 2 (4) None of these

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