Question

${\displaystyle \int \frac{\cos x+x\sin x}{x(x+\cos x)}}$dx = ?

विकल्प

• ${\displaystyle \log \left|\frac{1}{x+\cos x}\right|+c}$

• ${\displaystyle \log \left|\frac{x}{x+\cos x}\right|+c}$

• ${\displaystyle \log \left|\frac{1}{x+\sin x}\right|+c}$

• ${\displaystyle \log \left|\frac{x}{x+\sin x}\right|+c}$

Answer 1

${\displaystyle \log \left|\frac{x}{x+\cos x}\right|+c}$

Explanation:

${\displaystyle \int \frac{\cos x+x\sin x}{x(x+\cos x)}}$dx

${\displaystyle =\int \frac{(x+\cos x)-(x-x\sin x)}{x(x+\cos x)}}$dx

${\displaystyle =\int \frac{x+\cos x}{x(x+\cos x)}\text{dx}-\int \frac{x(1-\sin x)}{x(x+\cos x)}}$dx

${\displaystyle =\int \frac{\text{dx}}{x}-\int \frac{1-\sin x}{x+\cos x}\text{dx}}$

= log |x| - log |x + cos x| + c   ....${\displaystyle [\because \int \frac{\text{f'}\left(x\right)}{\text{f}\left(x\right)}\text{d}x=\log \left|\text{f}\left(x\right)\right|+c]}$

${\displaystyle \log \left|\frac{x}{x+\cos x}\right|+c}$