Question

Evaluate \[ \int \frac{x^2}{x^4 + 1} \, dx \]

Hint:

For integrals involving the form \( \frac{x^2}{x^4 + 1} \), try recognizing the derivative of the logarithmic function to simplify the integration.

Answer 1

Step 1: Simplifying the integrand.
The integral is: \[ \int \frac{x^2}{x^4 + 1} \, dx \] We can simplify this by recognizing that the denominator is \( x^4 + 1 \), which is the derivative of \( x^3 \), but we need to rewrite the numerator in a convenient form. Let us break \( x^2 \) as \( \frac{d}{dx}(x^3) \): \[ \int \frac{x^2}{x^4 + 1} \, dx = \frac{1}{2} \int \frac{d}{dx} \left( \ln(x^4 + 1) \right) \] Step 2: Performing the integration.
The integral of \( \frac{d}{dx} \left( \ln(x^4 + 1) \right) \) is simply: \[ \frac{1}{2} \ln(x^4 + 1) + C \] Step 3: Conclusion.
Thus, the solution is: \[ \frac{1}{2} \ln(x^4 + 1) + C \]