Question

If \(\int f(x)\,dx = g(x)\) then \(\int x^3f(x^2)\,dx\) is equal to

• A. \(\frac{1}{2}[x^2g(x^2)-\int g(x^2)\,d(x^2)]\)
• B. \(\frac{1}{2}[x^2[f(x)]^2-\int [g(x)]^2\,dx]\)
• C. \(\frac{1}{2}[x^2g(x)-\int g(x)\,d(x)]\)
• D. \(\frac{1}{2}[x^2g(x^2)+\int g(x^2)\,d(x^2)]\)

Hint:

Split x^2/((x^2+2)(x^2+5)) into partial fractions in x^2.

Answer 1

The Correct Option is A

Step 1: Idea:
Instead of solving for the partial fractions, we can differentiate the answer in option (D) and check that it returns the integrand.

Step 2: Direct check by differentiation:
Differentiate option (D): $-\frac{\sqrt2}{3}\cdot\frac{1/\sqrt2}{1 + x^2/2} + \frac{\sqrt5}{3}\cdot\frac{1/\sqrt5}{1 + x^2/5} = -\frac13\cdot\frac{2}{x^2 + 2} + \frac13\cdot\frac{5}{x^2 + 5}$.
Combine: $\frac13\cdot\frac{-2(x^2 + 5) + 5(x^2 + 2)}{(x^2 + 2)(x^2 + 5)} = \frac13\cdot\frac{3x^2}{(x^2+2)(x^2+5)} = \frac{x^2}{(x^2+2)(x^2+5)}$. This is the integrand, so (D) is correct.

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]