Question

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{2}}{\left(1+x^{3}\right)^{2}} d x $$

Short answer

The indefinite integral of \( \frac{x^{2}}{(1+x^{3})^{2}} \) is \( -\frac{1}{3(1+x^3)} + C \).

Step-by-step solution

Identifying the substitution

Looking at the integral, it is seen that the numerator is the derivative of the denominator. This makes it a candidate for substitution. Let's let \( u = 1 + x^3 \). The derivative of \( u \) with respect to \( x \) is \( du/dx = 3x^2 \), or \( dx = du / (3x^2) \). Write these substitutions down.

Substitute and Simplify

Replace \( dx \) and \( 1 + x^3 \) in the original integral with the substitutions determined in step 1. The integral becomes: \[ \int \frac{x^2}{u^2} \cdot \frac{du}{3x^2} \] The \( x^2 \) in the numerator and denominator cancel out to leave: \[ \int \frac{1}{3u^2} du \] Now the integral can be solved.

Solve the simplified integral

The integral is now much simpler and can be solved using basic integral rules: \[ \int \frac{1}{3u^2} du = -\frac{1}{3} \cdot \frac{1}{u} + C = -\frac{1}{3u} + C \] where \( C \) is the constant of integration.

Substitute the expression for u

Now substitute back the original expression for \( u \) (which was \( 1+x^3 \)) into the solution from step 3. This gives the following solution: \[ -\frac{1}{3(1+x^3)} + C \] This completes the integration.

Verify the result

Finally, to verify the result, differentiate the answer with respect to \( x \). If the derivative of the answer is equal to the original integrand, that confirms the solution: \[ \frac{d}{dx} \left(-\frac{1}{3(1+x^3)} + C\right) = \frac{x^2}{(1+x^3)^2} \] which is indeed the function we started with.