JEE Mains · Maths · STD 12 - 7.1 indefinite integral
For \({x^2} \ne n\pi + 1,\,n \in N\) (the set of natural numbers), the integral \(\int {x\sqrt {\frac{{2\,\sin \,\left( {{x^2} - 1} \right) - \sin \,2\,\left( {{x^2} - 1} \right)}}{{2\,\sin \,\left( {{x^2} - 1} \right) + \sin \,2\,\left( {{x^2} - 1} \right)}}} } \,dx\) is
- A \({\log _e}\,\left| {\frac{1}{2}\,{{\sec }^2}\,\left( {{x^2} - 1} \right)} \right| + c\)
- B \(\frac{1}{2}{\log _e}{\mkern 1mu} \left| {{\mkern 1mu} \sec {\mkern 1mu} \left( {{x^2} - 1} \right)} \right| + c\)
- C \(\frac{1}{2}{\log _e}\,\left| {\,{{\sec }^2}\,\left( {\frac{{{x^2} - 1}}{2}} \right)} \right| + c\)
- D \({\log _e}\,\left| {\,\sec \,\left( {\frac{{{x^2} - 1}}{2}} \right)} \right| + c\)
Answer & Solution
Correct Answer
(D) \({\log _e}\,\left| {\,\sec \,\left( {\frac{{{x^2} - 1}}{2}} \right)} \right| + c\)
Step-by-step Solution
Detailed explanation
\({\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)\left(1-\cos \left(x^{2}-1\right)\right)}{2 \sin \left(x^{2}-1\right)\left(1-\cos \left(x^{2}-1\right)\right)}}} \) \({=\int x \frac{\sin \left(\frac{x^{2}-1}{2}\right)}{\cos \left(\frac{x^{2}-1}{2}\right)} d x} \)…
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