Question

Evaluate \[ \int_{0}^{\pi/4} \frac{\cos x-\sin x}{9+5\sin 2x}\,dx. \]

• A. \[ \frac{1}{\sqrt5} \left( \tan^{-1}\sqrt{10} -\tan^{-1}\sqrt5 \right) \]
• B. \[ \frac{1}{2\sqrt5} \left( \tan^{-1}\frac{\sqrt{10}}{2} -\tan^{-1}\frac{\sqrt5}{2} \right) \]
• C. \[ \frac{1}{2\sqrt5} \left( \tan^{-1}\sqrt{10} +\tan^{-1}\sqrt5 \right) \]
• D. \[ \frac{1}{\sqrt5} \left( \tan^{-1}\sqrt{\frac52} +\tan^{-1}\frac{\sqrt5}{2} \right) \]

Hint:

Whenever the numerator contains \(\cos x-\sin x\), try the substitution \[ t=\sin x+\cos x, \] because \[ dt=(\cos x-\sin x)\,dx. \] Also remember \[ (\sin x+\cos x)^2=1+\sin 2x. \]

Answer 1

The Correct Option is B