$$ \int\left(4x+2\right) $$$$ \left(x^2+x\right)^7\differentialD x $$

To solve the integral 

\[
\int (4x + 2) (x^2 + x)^7 \, dx,
\]

we can use substitution. Let 

\[
u = x^2 + x.
\]

Then, the derivative $ du $ is given by 

\[
du = (2x + 1) \, dx.
\]

To express $ dx $ in terms of $ du $, we solve for $ dx $:

\[
dx = \frac{du}{2x + 1}.
\]

Next, we need to express $ 4x + 2 $ in terms of $ u $. Notice:

\[
4x + 2 = 4(x^2 + x) + 2 - 4x^2 = 4u + 2 - 4x^2.
\]

However, since $ x^2 = u - x $, this substitution becomes complex. Instead, let’s proceed by integrating by parts or using a simpler substitution.

Let's rewrite the integral as:

\[
\int (4x + 2) u^7 \, dx.
\]

Now, we can compute:

1. Substitute $ u $ and $ du $ back into the integral.
2. Simplify and integrate.

The new integral becomes:

\[
\int (4x + 2) u^7 \cdot \frac{du}{2x + 1}.
\]

However, the easier way is to notice that:

\[
\int (4x + 2) (x^2 + x)^7 \, dx = \int (4x + 2) \cdot u^7 \, dx,
\]

and integrate directly or recognize this form as a product of functions.

By applying integration by parts or recognizing the pattern of the product rule, we can solve this integral.

Would you like to explore a specific method or need a step-by-step guide on how to apply integration by parts here?

Discover how to solve the integral ∫(4x + 2)(x^2 + x)^7 dx using substitution and integration by parts. This detailed guide covers essential calculus concepts and methods, suitable for Grades 11-12.

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