Question

The integral of $f(x) = 1 + x^{2} + x^{4}$ with respect to $x^{2}$ is

• A. $x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + C$
• B. $\frac{x^{3}}{3} + \frac{x^{5}}{5} + C$
• C. $x^{2} + \frac{x^{4}}{4} + \frac{x^{6}}{6} + C$
• D. $x^{2} + \frac{x^{4}}{2} + \frac{x^{6}}{3} + C$

Hint:

"With respect to $x^2$" means $x^2$ is your variable. Don't multiply by $2x$!

Answer 1

The Correct Option is D

Step 1: Concept
This is an integration with respect to a function $u = x^2$.

Step 2: Meaning
Let $u = x^2$. The function becomes $f(u) = 1 + u + u^2$, and we are calculating $\int f(u) du$.

Step 3: Analysis
$\int (1 + u + u^2) du = u + \frac{u^2}{2} + \frac{u^3}{3} + C$.

Step 4: Conclusion
Substituting $u = x^2$ back in: $x^2 + \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} + C = x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C$.
Final Answer: (D)