91Ó°ÊÓ

The substitution method is a powerful technique that simplifies complex integrals. By recognizing certain parts of the integrand, you can transform it into a simpler expression. In this example, the integral is:\[ \int \frac{x^2 \, dx}{\sqrt{1-5x^3}} \]To use the substitution method, you first choose a substitution that makes the integration easier. Here, setting \( u = 1 - 5x^3 \) transforms the radical into a simple function, \( \sqrt{u} \). Then, differentiate \( u \) with respect to \( x \) to find \( du \):

• \( \frac{du}{dx} = -15x^2 \)
• so, \( du = -15x^2 \, dx \)

Now, express \( x^2 \, dx \) in terms of \( du \):- \( x^2 \, dx = -\frac{1}{15} \, du \)This allows you to rewrite the integral entirely in terms of \( u \), which simplifies the problem significantly.

After substitution, the integral becomes easier to solve. In our example, the integral simplifies to:\[ \int \frac{-1}{15} \frac{du}{\sqrt{u}} \]This is a prime candidate for the power rule for integration. The power rule states that:- \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)Here, you have \( u^{-1/2} \), so applying the power rule:- \(-\frac{1}{15} \int u^{-1/2} \, du \) Solving this:

• \(-\frac{1}{15} \times \frac{u^{1/2}}{1/2} = -\frac{2}{15} u^{1/2} + C \)

It's important to correctly handle the coefficients and exponents, as these play critical roles in the accuracy of integration.

After finding the antiderivative, it's essential to perform a differentiation check to confirm the solution matches the original integrand. This step ensures the integration process was correct.The result of the integral is:\[-\frac{2}{15} \sqrt{1-5x^3} + C \]Differentiate this expression with respect to \( x \). The differentiation will involve the chain rule:

• Differentiate \(-\frac{2}{15} (1-5x^3)^{1/2} \):
• \(-\frac{2}{15} \times \frac{-15x^2}{2(1-5x^3)^{1/2}} = \frac{x^2}{\sqrt{1-5x^3}} \)

The result matches the original integrand, \( \frac{x^2}{\sqrt{1-5x^3}} \), confirming that the integration was performed correctly. Using differentiation in this manner provides assurance that the integration technique has been applied properly.