To evaluate the integral of $x^7 e^{-2x^8}$, we will use substitution. Let's proceed step-by-step:

Step 1: Substitution

We want to simplify the expression $x^7 e^{-2x^8}$. A good substitution to try here is:

Let: $u = x^8$ Then, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 8x^7$ or equivalently: $du = 8x^7 dx$ This means we can rewrite the $x^7 dx$ term as: $x^7 dx = \frac{1}{8} du$

Step 2: Substituting into the Integral

Now, substitute $u = x^8$ and $x^7 dx = \frac{1}{8} du$ into the integral:

$\int x^7 e^{-2x^8} dx = \int e^{-2u} \cdot \frac{1}{8} du$

This simplifies to: $\frac{1}{8} \int e^{-2u} du$

Step 3: Integrating

The integral of $e^{-2u}$ is straightforward. The antiderivative of $e^{-2u}$ is: $\int e^{-2u} du = \frac{-1}{2} e^{-2u}$

Thus, the integral becomes: $\frac{1}{8} \cdot \left( \frac{-1}{2} e^{-2u} \right) = \frac{-1}{16} e^{-2u}$

Step 4: Substituting Back for $u$

Finally, recall that $u = x^8$. Substituting this back in, we get: $\frac{-1}{16} e^{-2x^8}$

Final Answer:

The integral of $x^7 e^{-2x^8}$ is: $\int x^7 e^{-2x^8} dx = \frac{-1}{16} e^{-2x^8} + C$ where $C$ is the constant of integration.

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Would you like any additional details or have any questions about this process?

Here are some related questions you might find interesting:

1. What happens if we change the exponent of $x$ in the integral?
2. How can substitution methods be applied to more complex integrals?
3. What is the general technique for evaluating integrals of exponential functions?
4. How does integration by parts differ from substitution?
5. Can we apply this substitution to other forms of exponential integrals?

Tip: When performing substitution, it's helpful to carefully choose a substitution that simplifies the expression and makes the integral easier to solve.