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JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Application +4 | -1
∫[x+x2+1ndx (n≠±1)\int\left[x+{\sqrt{x^2+1}}^n d x \,(n \neq \pm 1)\right. is
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Correct option: A
12[(x+x2+1)n+1n+1+(x+x2+1)n−1n−1]+c\frac{1}{2}\left[\frac{\left(x+\sqrt{x^2+1}\right)^{n+1}}{n+1}+\frac{\left(x+\sqrt{x^2+1}\right)^{n-1}}{n-1}\right]+c
Option A (12[(x+x2+1)n+1n+1+(x+x2+1)n−1n−1]+c\frac{1}{2}\left[\frac{\left(x+\sqrt{x^2+1}\right)^{n+1}}{n+1}+\frac{\left(x+\sqrt{x^2+1}\right)^{n-1}}{n-1}\right]+c) is correct because by substituting t=x+x2+1t=x+\sqrt{x^2+1} and simplifying, we can evaluate the integral.

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