Question

If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to_______

Answer 1

∫sec2x−1​dx=∫cos2x1−cos2x​​dx
=2​∫2cos2x−1​sinx​dx
put cosx=t⇒−sinxdx=dt
=−2​∫2t2−1​dt​
=−ln∣2​cosx+cos2x​∣+c
=−21​ln∣∣​2cos2x+cos2x+2cos2x​⋅2​cosx∣∣​+c
=−21​ln∣∣​cos2x+21​+cos2x​⋅1+cos2x​∣∣​+c
∵β=21​,α=−21​
⇒β−α=1
So, the correct answer is 1.