Evaluate the following integral.

Integral from nothing to nothing e Superscript 14 minus 15 x Baseline dx

It seems like you're asking for the evaluation of an integral involving the function \( e^{14 - 15x} \). I'll assume the limits of integration aren't specified since you've mentioned "from nothing to nothing." I'll work on the indefinite integral of the function \( e^{14 - 15x} \) with respect to \( x \).

### Solving the Indefinite Integral:

We need to evaluate:

\[
\int e^{14 - 15x} \, dx
\]

### Step 1: Substitution

Let \( u = 14 - 15x \). Then, the derivative of \( u \) with respect to \( x \) is:

\[
\frac{du}{dx} = -15 \quad \Rightarrow \quad dx = \frac{du}{-15}
\]

### Step 2: Substituting into the Integral

Substitute \( u \) and \( dx \) into the integral:

\[
\int e^{14 - 15x} \, dx = \int e^u \cdot \frac{du}{-15} = -\frac{1}{15} \int e^u \, du
\]

### Step 3: Evaluate the Integral

The integral of \( e^u \) is simply \( e^u \):

\[
-\frac{1}{15} \int e^u \, du = -\frac{1}{15} e^u + C
\]

### Step 4: Substitute Back \( u = 14 - 15x \)

Substituting \( u = 14 - 15x \) back into the expression:

\[
-\frac{1}{15} e^{14 - 15x} + C
\]

Thus, the indefinite integral is:

\[
\int e^{14 - 15x} \, dx = -\frac{1}{15} e^{14 - 15x} + C
\]

If you'd like to specify definite limits for the integral, I can further evaluate it.

Would you like more details on any step or a specific set of limits for this integral?

### Related Questions:
1. How do you find the definite integral of \( e^{14 - 15x} \) with specific limits?
2. How does substitution work in more complex integrals?
3. What are the applications of exponential integrals in physics?
4. Can we solve similar integrals without substitution?
5. How would a definite integral with bounds affect the final result?

### Tip:
Always check if substitution simplifies the expression before diving into complex integration techniques!

Learn how to evaluate the integral of e^(14 - 15x) with a step-by-step solution. This problem involves integration of exponential functions using substitution method. Suitable for advanced high school students or college calculus students.

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