Question:
The integral $\int \cos( \log \; x)dx$ is equal to : (where C is a constant of integration)
The integral $\int \cos( \log \; x)dx$ is equal to : (where C is a constant of integration)
Updated On: Apr 1, 2026
- $\frac{x}{2} \left[\sin\left(\log_{e} x\right)-\cos\left(\log_{e}x\right)+C\right]$
- $\frac{x}{2}\left[\cos\left(\log_{e}x\right) + \sin\left(\log_{e} x\right)\right]+C$
- $x [\cos (\log_e x) + \sin (\log_e x )] + C $
- $x [\cos (\log_e x) - \sin (\log_e x )] + C $
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The Correct Option is B
Solution and Explanation
$I = \int\cos \left(\ell n x \right)dx$
$ 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 $
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Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
Read More: Formation of a Differential Equation