The problem requires finding the indefinite integral of:

$\int (7x + 4)^5 \cdot 7 \, dx$

Solution

To solve this, we can use substitution. Let: $u = 7x + 4$

Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 7 \Rightarrow dx = \frac{du}{7}$

Substituting $u$ and $dx$ into the integral: $\int (7x + 4)^5 \cdot 7 \, dx = \int u^5 \cdot 7 \cdot \frac{du}{7}$

The $7$ terms cancel, so we have: $\int u^5 \, du$

Now, integrate $u^5$: $\int u^5 \, du = \frac{u^6}{6} + C$

Now, substitute back $u = 7x + 4$: $\frac{(7x + 4)^6}{6} + C$

Answer

The indefinite integral is: $\int (7x + 4)^5 \cdot 7 \, dx = \frac{(7x + 4)^6}{6} + C$

Verification by Differentiation

To verify, differentiate $\frac{(7x + 4)^6}{6} + C$ with respect to $x$:

1. Differentiate the outer part: $\frac{d}{dx} \left( \frac{(7x + 4)^6}{6} \right) = \frac{6(7x + 4)^5}{6} \cdot 7 = (7x + 4)^5 \cdot 7$

This matches the original integrand, confirming the solution.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How does substitution simplify complex integrals?
2. Why is it necessary to revert back to the original variable after substitution?
3. What is the importance of verifying an integral result through differentiation?
4. How would the approach change if the integral didn’t have the extra 7 factor?
5. Can you find the integral of $(ax + b)^n$ generally?

Tip: Always verify your integration by differentiating the result. It helps confirm that you have the correct solution.