91Ó°ÊÓ

The substitution method is a powerful technique in calculus used to simplify the process of integrating complex functions. This method involves transforming the original variable to a new one, making the integral easier to solve. In this case, the function given was \[ \int_{-1}^{1} x^{2}(x^{3}+1)^{4} \, dx \]Our goal is to simplify this by choosing an appropriate substitution. By letting \[ u = x^3 + 1 \]we effectively reduce the complexity of the expression inside the integral. Differentiating both sides with respect to \(x\) gives us:\[ \frac{du}{dx} = 3x^2 \] This allows us to express \(dx\) in terms of \(du\), resulting in\[ dx = \frac{du}{3x^2} \]The substitution transforms the original integral into a simpler one:\[ \int x^{2}(u)^{4} \left(\frac{du}{3x^{2}}\right) = \frac{1}{3} \int u^{4} \, du \]This new integral is much more straightforward to solve using basic integration rules.

• Choose a substitution to simplify the integral.
• Differentiate to find \(du\) and substitute \(dx\).
• Transform the original integral to a simpler one.

When using the substitution method, it’s essential to adjust the integration limits to correspond to the new variable. Initially, the limits pertain to the original variable \(x\), but after substituting, the limits must match the new variable \(u\). Initially, the limits were \(x = -1\) and \(x = 1\). We update them using the substitution \(u = x^3 + 1\):

• For \(x = -1\): \[ u = (-1)^3 + 1 = 0 \]
• For \(x = 1\): \[ u = (1)^3 + 1 = 2 \]

Thus, the transformed integral becomes:\[ \frac{1}{3} \int_{0}^{2} u^{4} \, du \]Updating the integration limits allows us to integrate within the scope of the new variable correctly. Always remember to reflect any substitution in the limits.

The power rule is a fundamental technique used to evaluate integrals where the function is a simple power of the variable. It states that the integral of \(u^n\) with respect to \(u\) is:\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]where \(n\) is a constant. In our integral, this power rule applies directly to integrate:\[ \int u^{4} \, du \]Following the power rule, the antiderivative becomes:\[ \frac{u^{5}}{5} \]Using the power rule is straightforward, as it only requires increasing the exponent by one and dividing by the new exponent. Since it's a definite integral, we eliminate the constant \(C\) and evaluate from the updated limits \(u=0\) to \(u=2\):\[ \frac{1}{3} \left[ \frac{u^{5}}{5} \right]_{0}^{2} = \frac{1}{3} \left[ \frac{32}{5} - 0 \right] = \frac{32}{15} \]By applying the power rule, we efficiently solve the integral and obtain the final result. This step showcases the power of simple rules in handling otherwise complex integrals. Remember to apply these steps methodically.