Question

Evaluate $\int \frac{\sqrt{58}dx}{7\cos x+3\sin x}$
(where $C$ is constant of integration)

• A. $\ln \left|\tan (\frac{x}{2}+\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$
• B. $\ln \left|\tan (\frac{x}{2}-\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$
• C. $-\ln \left|\tan (\frac{x}{2}+\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$
• D. $-\ln \left|\tan (\frac{x}{2}-\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$

Answer 1

The correct option is A $\ln \left|\tan (\frac{x}{2}+\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$

Let $I=\int \frac{\sqrt{58}dx}{7\cos x+3\sin x}$
$\Rightarrow I=\int \frac{dx}{\frac{7}{\sqrt{5}8}\cos x+\frac{3}{\sqrt{5}8}\sin x}$

$\Rightarrow I=\int \frac{dx}{\sin \theta \cdot \cos x+\cos \theta \sin x}$ where,
$\left[\begin{array}{c}\frac{7}{\sqrt{58}}=\sin \theta \\ \frac{3}{\sqrt{58}}=\cos \theta \\ \tan \theta =7/3\end{array}\right]$

$\Rightarrow I=\int \frac{dx}{\sin (x+\theta )}=\int cosec(x+\theta )dx$
$\Rightarrow I=\ln \left|\tan (\frac{x}{2}+\frac{\theta }{2})\right|+C$
$\therefore I=\ln \left|\tan (\frac{x}{2}+\frac{1}{2}{\tan }^{-1}\frac{7}{3})\right|+C$