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: Understanding the Concept:
The phrase "with respect to \(x^2\)" means that \(x^2\) is our variable of integration, often denoted by \(u\). We substitute \(u = x^2\) and integrate the resulting polynomial in terms of \(u\).
Step 2: Key Formula or Approach:
1. Let \(u = x^2\).
2. Rewrite the function: \(f(u) = 1 + u + u^2\).
3. Use \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\).
Step 3: Detailed Explanation:
We are finding \(\int (1 + x^2 + x^4) \, d(x^2)\). Let \(u = x^2\). The integral becomes: \[ \int (1 + u + u^2) \, du \] Integrate term by term: \[ = u + \frac{u^2}{2} + \frac{u^3}{3} + C \] Substitute back \(u = x^2\): \[ = x^2 + \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} + C = x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C \]
Step 4: Final Answer:
The integral is \( x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C \).