Evaluating Integrals: A Step-by-Step Guide with an Example

When dealing with integrals, it's important to use the correct terminology. One does not solve integrals, but rather evaluates them. In this article, we will walk through the process of evaluating a specific integral, providing a detailed solution and explaining the steps involved.

Consider the integral $$int_{0}^{1} frac{x^2}{x^3 - 1 sqrt{1 - x^6}}, dx$$, where the variable is first transformed to simplify the expression.

Step 1: Substitution and Simplification

The integral is transformed using the substitution: $tan t x^3$. This substitution is chosen because it simplifies the trigonometric expression.

First, rewrite the integral with the substitution:

$$int_{0}^{1} frac{x^2}{x^3 - 1 sqrt{1 - x^6}}, dx int_{0}^{1} frac{x^2}{(tan t) - 1 sqrt{1 - (tan t)^2}}frac{sec^2 t, dt}{3x^2}$$

Note that $x^2, dx frac{sec^2 t, dt}{3}$ and the new limits of integration, which are from $0$ to $frac{pi}{4}$, need to be converted correspondingly.

The integral now becomes:

$$frac{1}{3} int_{0}^{frac{pi}{4}} frac{sec^2 t}{tan t - 1 sqrt{1 - tan^2 t}}, dt frac{1}{3} int_{0}^{frac{pi}{4}} frac{sec t}{tan t - 1}, dt$$

Further simplification involves recognizing that $1 - tan^2 t cos^2 t - sin^2 t$. Thus, $sqrt{1 - tan^2 t} frac{1}{cos t}$, simplifying the integral further:

$$frac{1}{3} int_{0}^{frac{pi}{4}} frac{sec t}{tan t - 1}, dt frac{1}{3} int_{0}^{frac{pi}{4}} frac{mathrm{d}t}{sin t cos t} frac{1}{3} int_{0}^{frac{pi}{4}} frac{mathrm{d}t}{sqrt{2} cosleft(t frac{pi}{4}right)}$$

This integral can be solved using the substitution $t frac{pi}{4} u$, which leads to:

$$frac{1}{3sqrt{2}} int_{0}^{frac{pi}{4}} secleft(t frac{pi}{4}right), dt frac{1}{3sqrt{2}} left[lnleft|secleft(t frac{pi}{4}right) tanleft(t frac{pi}{4}right)right|right]_0^{frac{pi}{4}}$$

Substituting the limits of integration:

$$frac{1}{3sqrt{2}} left[lnleft|sec 0 - tan 0right| - lnleft|secleft(-frac{pi}{4}right) - tanleft(-frac{pi}{4}right)right|right] frac{1}{3sqrt{2}} left[0 - lnleft|frac{sqrt{2}}{1 - 1}right|right] frac{-1}{3sqrt{2}} lnleft(sqrt{2} - 1right) approx 0.20774$$

Step-by-Step Solution with WolframAlpha

To evaluate this integral, you can use WolframAlpha. Enter the integral in the following format:

$$text{integrate } frac{x^2}{x^3 - 1sqrt{1 - x^6}}, dx text{ from } 0 text{ to } 1$$

Click on the Step-by-Step Solution button to see how the integral is evaluated. This step-by-step approach can help you understand the process and verify your solution.

For a full solution, you may need access to WolframAlpha Pro, but the website often provides a good start. Additionally, you can find solutions involving trigonometric and hyperbolic trigonometric functions somewhere along the way.

In conclusion, the integral evaluates to approximately 0.20774, demonstrating the power of substitution and trigonometric identities in solving complex integrals.