Question

Let x²¹ np – 1, n Î N then $\int_{}^{}{x\sqrt{\frac{2\sin(x^{2} + 1)–\sin 2(x^{2} + 1)}{2\sin(x^{2} + 1) + \sin 2(x^{2} + 1)}}}$dx equals

• A. ln $\left| \frac{1}{2}\sec(x^{2} + 1) \right|$ + C
• B. ln $\left| \sec{}\left( \frac{x^{2} + 1}{2} \right) \right|$ + C
• C. $\frac{1}{2}$ln | sec(x² + 1) | + C
• D. $\frac{1}{2}$ ln $\left| \frac{2}{\sec(x^{2} + 1)} \right|$ + C

Answer 1

Answer: (B)

ln $\left| \sec{}\left( \frac{x^{2} + 1}{2} \right) \right|$ + C

Answer 2

$\frac{1}{2}$ dx

Put x² + 1 = t ̃ 2xdx = dt

= $\frac{1}{2}$ $\int_{}^{}\sqrt{\frac{2\sin t–\sin 2t}{2\sin t + \sin 2t}}$ dt

= $\frac{1}{2}$ $\int_{}^{}\sqrt{\frac{2–2\cos t}{2 + 2\cos t}}$ dt

= $\frac{1}{2}$ $\int_{}^{}{\tan\frac{t}{2}dt}$ = $\frac{\frac{1}{2}\mathcal{l}n|\sec t/2| + C}{1/2}$

= ln $\left| \sec\left( \frac{x^{2} + 1}{2} \right) \right|$ + C