Indefinite Integration
Integration by Substitution
JEE Main 2022
Grade 12
Question:
$\int \frac{x^3}{\sqrt{1 + x^2}} \, dx$ equals:
(1) $\frac{1}{3}(x^2 - 2)\sqrt{1 + x^2} + C$
(2) $\frac{1}{3}(x^2 + 2)\sqrt{1 + x^2} + C$
(3) $(x^2 - 2)\sqrt{1 + x^2} + C$
(4) $\frac{1}{3}x^2 \sqrt{1 + x^2} + C$
Step-by-Step Solution
Key Concept: Substitute $t = 1 + x^2$ (so $x^2 = t - 1$ and $x \, dx = \frac{1}{2} dt$) to rewrite the integrand entirely in terms of $t$ and $\sqrt{t}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)