Understanding Integration with Respect to a Function

The question asks for the integral of the function \(f(x) = 1 + x^2 + x^4\) with respect to \(x^2\). This is different from integrating with respect to \(x\). When we integrate with respect to a function, say \(g(x)\), we are essentially finding \(\int f(x) \, d(g(x))\).

Applying Substitution for Integration

In this case, the integration is with respect to \(x^2\). Let's use a substitution to make this clearer. Let \(u = x^2\). The problem then becomes finding the integral of the function \(f(x)\) with respect to \(u\).

We need to express \(f(x) = 1 + x^2 + x^4\) in terms of \(u\). Since \(u = x^2\), we have:

• The constant term is \(1\).
• The second term is \(x^2\), which is equal to \(u\).
• The third term is \(x^4 = (x^2)^2\), which is equal to \(u^2\).

So, the function \(f(x)\) expressed in terms of \(u\) is \(1 + u + u^2\).

Performing the Integration

Now, we need to integrate \(1 + u + u^2\) with respect to \(u\). The integral is:

We can integrate each term separately:

• Integral of \(1\) with respect to \(u\) is \(\int 1 \, du = u\).
• Integral of \(u\) with respect to \(u\) is \(\int u \, du = \frac{u^{1+1}}{1+1} = \frac{u^2}{2}\).
• Integral of \(u^2\) with respect to \(u\) is \(\int u^2 \, du = \frac{u^{2+1}}{2+1} = \frac{u^3}{3}\).

Combining these terms and adding the constant of integration \(C\), we get:

Substituting Back to Original Variable

Finally, we substitute back \(u = x^2\) into the result:

Comparing with Options

Let's compare our result with the given options:

1. \(\rm x + \frac{x^3}{3}+\frac{x^5}{5}+C\)
2. \(\rm \frac{x^3}{3}+\frac{x^5}{5}+C\)
3. \(\rm x^2 + \frac{x^4}{4}+\frac{x^6}{6}+C\)
4. \(\rm x^2 + \frac{x^4}{2}+\frac{x^6}{3}+C\)

Our result, \(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C\), matches option 4.

Therefore, the integral of \(f(x) = 1 + x^2 + x^4\) with respect to \(x^2\) is \(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + C\).