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Let x^2 \neq n \pi-1, n \in N then

\int x \sqrt{\frac{2 \sin \left(x^2+1\right)-\sin 2\left(x^2+1\right)}{2 \sin \left(x^2+1\right)+\sin 2\left(x^2+1\right)}} d x is equal to 

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

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


Answers (1)

best_answer

\begin{aligned} & I=\frac{1}{2} \int 2 x \sqrt{\frac{2 \sin \left(x^2+1\right)-\sin 2\left(x^2+1\right)}{2 \sin \left(x^2+1\right)+\sin 2\left(x^2+1\right)}} d x \\ \\& x^2+1=t \Rightarrow 2 x d x=d t \\ \\& I=\frac{1}{2} \int \sqrt{\frac{2 \sin t-\sin 2 t}{2 \sin t+\sin 2 t}} d t \\ \\& =\frac{1}{2} \int \sqrt{\frac{2-2 \cos t}{2+2 \cos t}} d t \\ & \end{aligned}

\begin{aligned} & =\frac{1}{2} \int \tan \frac{t}{2} d t \\ \\& =\frac{1}{2} \frac{\ln \left|\sec \frac{t}{2}\right|}{\frac{1}{2}}+c \\ \\& =\ln \left|\sec \left(\frac{x^2+1}{2}\right)\right|+c \end{aligned}

Posted by

Gautam harsolia

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