If the anti-derivative of $\frac{x^3}{\sqrt{4+2x^2}}$ which passes through $(1, 2)$ is $\frac{1}{m}\left(1+2x^2\right)^{1/2}\left(x^2-1\right)+c$. Then:
Step-by-Step Solution
Key Concept: Substituting the expression under the radical as a new variable eliminates the square root and converts the problem into a rational function.
For $\int \frac{x^3}{\sqrt{1+2x^2}}dx$, we substitute $1 + 2x^2 = t^2$, giving $4xdx = 2tdt$ and $x^2 = \frac{t^2-1}{2}$. The integral becomes $\frac{1}{2}\int \frac{(t^2-1)udt}{2t}$, which after simplification yields $\frac{t}{12}(t^2-3) + c = \frac{\sqrt{1+2x^2}(2x^2-1)}{6} + c$, so $m = 6$.
Correct Answer: 2,4