Consider the following integral. Find a change of variables to rewrite the integrand as \( u^{2} d u \). \( \left.x^{2}-3\right)^{2}(2 x) d x \) \( u=\square \) \( d u=\square \) Evaluate the given integral. (Remember the constant of integration.) du R

To evaluate the integral \[ \int \left(x^{2} - 3\right)^{2} (2x) \, dx, \] we can use the method of **substitution** (also known as **change of variables**). Here's a step-by-step guide: ### Step 1: Choose a Substitution Let’s choose: \[ u = x^{2} - 3 \] This substitution simplifies the expression inside the integral. ### Step 2: Differentiate to Find \( du \) Differentiate both sides with respect to \( x \): \[ \frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx \] Notice that \( 2x \, dx \) is present in the original integral, which matches \( du \). ### Step 3: Rewrite the Integral in Terms of \( u \) and \( du \) Substitute \( u \) and \( du \) into the integral: \[ \int \left(x^{2} - 3\right)^{2} (2x) \, dx = \int u^{2} \, du \] ### Step 4: Integrate with Respect to \( u \) Now, integrate \( u^{2} \) with respect to \( u \): \[ \int u^{2} \, du = \frac{u^{3}}{3} + C \] where \( C \) is the constant of integration. ### Step 5: Substitute Back \( u = x^{2} - 3 \) Finally, replace \( u \) with the original substitution to express the answer in terms of \( x \): \[ \frac{u^{3}}{3} + C = \frac{(x^{2} - 3)^{3}}{3} + C \] ### Final Answer \[ \int \left(x^{2} - 3\right)^{2} (2x) \, dx = \frac{(x^{2} - 3)^{3}}{3} + C \] **Summary of Substitution:** \[ u = x^{2} - 3 \] \[ du = 2x \, dx \] And the evaluated integral is: \[ \frac{(x^{2} - 3)^{3}}{3} + C \]