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

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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:
$$
\frac{(x^{2} - 3)^{3}}{3} + C
$$
Let $$ u = x^{2} - 3 $$, then $$ du = 2x \, dx $$. The integral becomes $$ \int u^{2} \, du = \frac{u^{3}}{3} + C = \frac{(x^{2} - 3)^{3}}{3} + C $$.

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

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