Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{-1}^{1} x\left(x^{2}+1\right)^{3} d x $$

Short answer

The value of the definite integral is 2

Step-by-step solution

Identify the function and its derivative

In the given expression \(x(x^2+1)^3\), we can see that the derivative of \((x^2+1)\) is \(2x\). The function can be rewritten in a form suitable for substitution: \( \frac{1}{2} * 2x(x^2+1)^3\) .

Perform the substitution

Let's set \(u = x^2+1\). The derivative of \(u\) is \(du = 2x dx\). The original expression in terms of \(u\) becomes \( \frac{1}{2} * u^3 du\) .

Evaluate the new integral

Now compute the integral: \(\frac{1}{2} * \int_{0}^{2} u^3 du\). By power rule of integration, that equals to \( \frac{1}{2} * (\frac{u^4}{4})\) evaluated from 0 to 2, which equals \( \frac{1}{2} * ( (\frac{2^4}{4}) - (\frac{0^4}{4}) ) = \frac{1}{2} * 4 = 2 \)

Verify with a graphing utility

To confirm this solution, plot the function \(x(x^2+1)^3\) on a graphing utility and calculate the area under the curve from -1 to 1. The result should approximately equal to the calculated integral.