We are tasked with evaluating the integral:

$\int x(x^2 - 1)^5 \, dx$

Step 1: Apply u-substitution

Let:

$u = x^2 - 1$

Differentiate both sides with respect to $x$:

$\frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx$

Solve for $x \, dx$:

$x \, dx = \frac{1}{2} \, du$

Step 2: Substitute into the integral

Substituting $u = x^2 - 1$ and $x \, dx = \frac{1}{2} \, du$, the integral becomes:

$\int x(x^2 - 1)^5 \, dx = \int (u^5) \cdot \frac{1}{2} \, du$

Simplify:

$\frac{1}{2} \int u^5 \, du$

Step 3: Integrate

Use the power rule for integration:

$\int u^5 \, du = \frac{u^6}{6}$

Thus, the integral becomes:

$\frac{1}{2} \cdot \frac{u^6}{6} = \frac{u^6}{12}$

Step 4: Substitute back $u = x^2 - 1$

$\frac{(x^2 - 1)^6}{12} + C$

Final Answer:

$\int x(x^2 - 1)^5 \, dx = \frac{(x^2 - 1)^6}{12} + C$

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Would you like a more detailed breakdown or clarification on any step?

Related Questions:

1. How would you integrate $x(x^2 + 1)^5 \, dx$ using $u$-substitution?
2. Can you solve $\int x^2(x^2 - 1)^5 \, dx$ with substitution?
3. How does $u$-substitution relate to the chain rule in differentiation?
4. What if the integrand were $x^3(x^2 - 1)^5$? How would the process change?
5. How do you verify the result by differentiating $\frac{(x^2 - 1)^6}{12}$?

Tip:

When using $u$-substitution, always check if part of the integrand is the derivative of another part. This makes substitution much smoother!