Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{-1}^{1} x^{2}\left(x^{3}+1\right)^{4} d x$$

Step-by-Step Solution

Answer

The definite integral \(\int_{-1}^{1} x^{2}\left(x^{3}+1\right)^{4} d x\) can be evaluated using substitution method, where \(u= x^3+1\). After updating the limits and simplifying the integral, we get \(\frac{1}{3}\int_{0}^{2} u^{4} d u\). Evaluating this integral gives \(\frac{32}{15}\) as the final answer.

Step 1: Choose a substitution

Let's choose a substitution in order to simplify the expression inside the integral. We can let: $$u = x^3 + 1$$ Now, we need to find out the differential, \(du\). Differentiating both sides with respect to \(x\) gives: $$\frac{d u}{d x} = 3x^2$$ Then, solve for \(dx\): $$d x=\frac{d u}{3 x^{2}}$$

Step 2: Replace the variables in the integral

We can now replace \(x^3 + 1\) with \(u\), and \(dx\) with \(\frac{d u}{3 x^{2}}\). Our integral becomes: $$\int x^{2}\left(u\right)^{4}\left(\frac{d u}{3 x^{2}}\right)$$ Now, simplify the integral: $$\frac{1}{3}\int u^{4} d u$$

Step 3: Update integration limits

Since we changed the variable to \(u\), we should also update the integration limits. Our original integral had the limits \(x = -1\) and \(x = 1\). Using the substitution \(u = x^3 + 1\), we can rewrite the limits in terms of \(u\): For \(x = -1\), \(u = (-1)^3 + 1 = 0\) For \(x = 1\), \(u = (1)^3 + 1 = 2\) Our new integral with the updated integration limits and variables is given by: $$\frac{1}{3}\int_{0}^{2} u^{4} d u$$

Step 4: Evaluate the integral

Using the power rule for integration, we can evaluate the definite integral. The power rule states that the integral of \(u^n\) with respect to \(u\) is \(\frac{u^{n+1}}{n+1} + C\), where \(n\) is a constant: $$\frac{1}{3}\int_{0}^{2} u^{4} d u = \frac{1}{3}\left[\frac{u^{5}}{5}\right]_{0}^{2}$$ Evaluate the integral at the limits and compute the difference: $$\frac{1}{15}(2^5 - 0^5) = \frac{1}{15}(32) = \frac{32}{15}$$

Step 5: Write the final answer

The definite integral is equal to: $$\int_{-1}^{1} x^{2}\left(x^{3}+1\right)^{4} d x = \frac{32}{15}$$

Key Concepts

Substitution Method

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.

Integration Limits

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.

Power Rule for Integration

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.