First slide

Methods of integration

Question

Let $x^{2} \neq nπ - 1, n \in N$ .The value of $\int x \sqrt{\frac{2 \sin (x^{2} + 1) - \sin 2 (x^{2} + 1)}{2 \sin (x^{2} + 1) + \sin 2 (x^{2} + 1)}} dx$ is equal to

Moderate

A

$\log |\frac{1}{2} \sec (x^{2} + 1)| + C$
B

$\log |\sec (\frac{x^{2} + 1}{2})| + C$
C

$\frac{1}{2} \log |\sec (x^{2} + 1)| + C$
D

None of the above

Solution

We have, $\int x \sqrt{\frac{2 \sin (x^{2} + 1) - \sin 2 (x^{2} + 1)}{2 \sin (x^{2} + 1) + \sin 2 (x^{2} + 1)}} dx$
$\begin{array}{l} = \int x \sqrt{\frac{2 \sin (x^{2} + 1) - 2 \sin (x^{2} + 1) \cdot \cos (x^{2} + 1)}{2 \sin (x^{2} + 1) + 2 \sin (x^{2} + 1) \cdot \cos (x^{2} + 1)}} dx \\ = \int x \sqrt{\frac{1 - \cos (x^{2} + 1)}{1 + \cos (x^{2} + 1)}} dx = \int xtan (\frac{x^{2} + 1}{2}) dx \\ = \int \tan (\frac{x^{2} + 1}{2}) d (\frac{x^{2} + 1}{2}) \\ = \log |\sec (\frac{x^{2} + 1}{2})| + C \end{array}$

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