Question

The integral $\int \cos( \log \; x)dx$ is equal to : (where C is a constant of integration)

• A. $\frac{x}{2} \left[\sin\left(\log_{e} x\right)-\cos\left(\log_{e}x\right)+C\right]$
• B. $\frac{x}{2}\left[\cos\left(\log_{e}x\right) + \sin\left(\log_{e} x\right)\right]+C$
• C. $x [\cos (\log_e x) + \sin (\log_e x )] + C$
• D. $x [\cos (\log_e x) - \sin (\log_e x )] + C$

Answer 1

Answer: (B)

$\frac{x}{2}\left[\cos\left(\log_{e}x\right) + \sin\left(\log_{e} x\right)\right]+C$

Answer 2

$I = \int\cos \left(\ell n x \right)dx$ $I = \cos\left(ln x\right) .x + \int \sin\left(\ell nx\right)dx$ $\cos\left(\ell n x\right)x+\left[\sin\left(\ell nx\right). x -\int\cos\left(\ell n x\right) dx\right]$ $I = \frac{x}{2} \left[\sin\left(\ell nx\right)+\cos\left(\ell nx\right)\right]+C$