JEE MainMathematicsModerateDefinite Integration

Value
of $\int_{0}^{\pi/4}\mspace{2mu}\mathcal{l}n(1 + cos4x)$ dx is.

Show answer and solution

\int_{0}^{\pi/4}\mspace{2mu}\mathcal{l}n(1 + cos4x)dx = \int_{0}^{\pi/4}\mspace{2mu}\mathcal{l}n\left( 2\cos^{2}2x \right)dx = \frac{1}{2}\int_{0}^{\pi/2}\mspace{2mu}\mathcal{l}n\left( 2\cos^{2}x \right)dx

= \frac{1}{2}\int_{0}^{\pi/2}\mspace{2mu}\left( \mathcal{l}n2 + \mathcal{l}n\cos^{2}x \right)dx

= \frac{\pi}{4}\mathcal{l}n2 + \int_{0}^{\pi/2}\mspace{2mu}\mathcal{l}ncosxdx = - \frac{\pi}{4}\mathcal{l}n2

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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