Question

$\int \frac{1}{\sqrt{{\sec }^{2}x+{\tan }^{2}x}}dx=\log |f\left(x\right)+\sqrt{g\left(x\right)}|+C$, then which of the following is/are correct?

• A. $f\left(x\right)=\sin x$
• B. $f\left(x\right)=\cos x$
• C. $g\left(x\right)=1+{\sin }^{2}x$
• D. $g\left(x\right)=1+{\cos }^{2}x$

Answer 1

Answer: (A), (C)

$g\left(x\right)=1+{\sin }^{2}x$

$\int \frac{\cos x}{\sqrt{1+{\sin }^{2}x}}dx\Rightarrow \sin x=t\Rightarrow \cos xdx=dt=\int \frac{dt}{\sqrt{1+t^2}}=\log |t+\sqrt{1+t^2}|+C=\log |\sin x+\sqrt{1+{\sin }^{2}x}|+C$