The problem in the image is to integrate the following expression:

$\int x (x^2 + 3)^5 \, dx$

Let's solve this step-by-step.

Step 1: Use Substitution

Let: $u = x^2 + 3$

Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 2x \Rightarrow du = 2x \, dx \Rightarrow \frac{du}{2} = x \, dx$

Step 2: Substitute in Terms of $u$

Now, rewrite the integral in terms of $u$: $\int x (x^2 + 3)^5 \, dx = \int (x^2 + 3)^5 \cdot x \, dx = \int u^5 \cdot \frac{du}{2}$

This simplifies to: $\frac{1}{2} \int u^5 \, du$

Step 3: Integrate with Respect to $u$

Now, integrate $u^5$ with respect to $u$: $\frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^6}{6} = \frac{u^6}{12}$

Step 4: Substitute Back in Terms of $x$

Now, replace $u$ with $x^2 + 3$: $\frac{u^6}{12} = \frac{(x^2 + 3)^6}{12}$

Final Answer

So, the integral is: $\int x (x^2 + 3)^5 \, dx = \frac{(x^2 + 3)^6}{12} + C$ where $C$ is the constant of integration.

Would you like further details or have any questions on the steps?

Related Questions:

1. What is the substitution method in integration?
2. How does substitution simplify complex integrals?
3. What are common integration techniques besides substitution?
4. Why is it necessary to include the constant of integration?
5. Can substitution be used for definite integrals as well?

Tip: When using substitution, always express $dx$ in terms of $du$ to simplify the integral correctly.