To solve the integral

$\int_{1}^{2} \frac{x + 1}{(x^2 + 2x)^3} \, dx,$

we will use substitution. Let’s define a new variable $u$ to simplify the expression.

1. Substitution:

   Let $u = x^2 + 2x$. Then, we need to find $du$:

   $du = (2x + 2) \, dx = 2(x + 1) \, dx \implies dx = \frac{du}{2(x + 1)}.$

2. Rewrite the Integral:

   Substitute $u$ and $dx$ into the integral:

   $\int_{1}^{2} \frac{x + 1}{u^3} \cdot \frac{du}{2(x + 1)} = \int_{u(1)}^{u(2)} \frac{1}{2u^3} \, du.$

   Since $u = x^2 + 2x$:

   • When $x = 1$: $u(1) = 1^2 + 2 \cdot 1 = 3$.
   • When $x = 2$: $u(2) = 2^2 + 2 \cdot 2 = 8$.

   Therefore, the integral bounds change from 1 to 3 and 2 to 8 respectively.

3. Simplify the Integral:

   Now the integral is:

   $\frac{1}{2} \int_{3}^{8} u^{-3} \, du.$

4. Integrate:

   Use the power rule for integration:

   $\int u^{-3} \, du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}.$

   Therefore:

   [ \frac{1}{2} \left[-\frac{1}{2u^2}\right]{3}^{8} = -\frac{1}{4} \left[\frac{1}{u^2}\right]{3}^{8}. ]

5. Evaluate the Bounds:

   Substitute the bounds into the expression:

   $-\frac{1}{4} \left( \frac{1}{8^2} - \frac{1}{3^2} \right) = -\frac{1}{4} \left( \frac{1}{64} - \frac{1}{9} \right).$

   Find a common denominator to simplify:

   $\frac{1}{64} = \frac{1}{64}, \quad \frac{1}{9} = \frac{64}{576}.$

   $\frac{1}{64} - \frac{64}{576} = \frac{9}{576} - \frac{64}{576} = \frac{9 - 64}{576} = \frac{-55}{576}.$

   Therefore:

   $-\frac{1}{4} \cdot \frac{-55}{576} = \frac{55}{2304}.$

So, the value of the integral is

$\frac{55}{2304}.$

Do you want any more details or have any questions?

Here are five related questions you can ask:

1. How do you find the antiderivative of a polynomial function?
2. Can you show an example of integration by parts?
3. How do you solve an integral using partial fractions?
4. What is the method of trigonometric substitution in integration?
5. How do you apply the Fundamental Theorem of Calculus?

Tip: When performing substitution in integrals, always adjust the limits of integration to the new variable to simplify the computation.