91Ó°ÊÓ

The substitution method is a powerful tool for simplifying integrals. It involves changing variables to make the integral easier to evaluate. To apply substitution, follow these steps:

• Identify a part of the integrand to substitute with a new variable, often denoted as u.
• Rewrite the integral in terms of this new variable.
• Integrate with respect to u.
• Finally, substitute back the original variable to get the result.

In the given problem, we use the substitution u = x + 2. This simplifies the integral because du = dx, making the new integral \(\frac{du}{u^3}\). The complicated function in the integrand becomes easier to handle.

The power rule for integration is a fundamental rule that helps in integrating functions of the form \(x^n\). According to this rule:
\[ \int u^n du = \frac{u^{n+1}}{n+1}, \text{ for } n eq -1\] When using the power rule, it's critical to adjust the function so that it's in the form \(u^n\). In our problem, after substitution, we get \(\frac{du}{u^3}\), which is the same as integrating \(u^{-3}\). Using the power rule, we have:
\[ \int u^{-3} du = \frac{u^{-2}}{-2} = -\frac{1}{2} u^{-2} \]
This rule simplifies the process of finding the antiderivative.

The constant of integration is an essential part of indefinite integrals. When you integrate a function, you add an arbitrary constant \(C\) to the result. This is because integration is the inverse operation of differentiation, and the derivative of a constant is zero. Therefore, any constant could have been present in the original function before differentiation.
In our problem, after integrating and substituting back, we obtain: \[-\frac{1}{2(u^2)} = -\frac{1}{2(x+2)^2}\text{ and then add the constant of integration } C \] Therefore, the final solution is: \[ \int \frac{dx}{(x+2)^3} = - \frac{1}{2 (x+2)^2} + C \]
Always remember to include \(C\) when dealing with indefinite integrals to ensure your solution is complete.