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JEE MainMathematicsModerateIndefinite integration

For x2≠nπ+1,n∈Nx^{2} \neq n\pi + 1,n \in N(the set of natural numbers),
theintegral∫x2sin(x2−1)−sin2(x2−1)2sin(x2−1)+sin2(x2−1)dx\int x\sqrt{\frac{2sin\left( x^{2} - 1 \right) - sin2\left( x^{2} - 1 \right)}{2sin\left( x^{2} - 1 \right) + sin2\left( x^{2} - 1 \right)}}dxis
equal

Show answer and solution
∫x(1−cos(x2−1))(1+cos(x2−1))dx\int x\sqrt{\frac{\left( 1 - cos\left( x^{2} - 1 \right) \right)}{\left( 1 + cos\left( x^{2} - 1 \right) \right)}}dx
=∫xsin(x2−12)cos(x2−12)dx= \int x\frac{sin\left( \frac{x^{2} - 1}{2} \right)}{cos\left( \frac{x^{2} - 1}{2} \right)}dx
=∫x⋅tan(x2−12)dx= \int x \cdot tan\left( \frac{x^{2} - 1}{2} \right)dx
x2−12=t⇒2xdx=2dt\frac{x^{2} - 1}{2} = t \Rightarrow 2xdx = 2dt
=∫tantdt=lnsect+c=lnsec(x2−12)+c= \int tantdt = \mathcal{l}nsect + c = \mathcal{l}nsec\left( \frac{x^{2} - 1}{2} \right) + c

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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