To solve the integral 

\[
\int a \cdot e^{-b(x + t)} \, dx,
\]

let's break this down step-by-step:

### 1. **Identify constants and variables:**

- \( a \) and \( b \) are constants.
- \( t \) is a constant.
- \( x \) is the variable of integration.

### 2. **Simplify the integrand:**

The integrand \( a \cdot e^{-b(x + t)} \) can be rewritten using the properties of exponents:

\[
a \cdot e^{-b(x + t)} = a \cdot e^{-bx} \cdot e^{-bt}.
\]

Since \( e^{-bt} \) is a constant with respect to \( x \), we can factor it out of the integral:

\[
\int a \cdot e^{-b(x + t)} \, dx = a \cdot e^{-bt} \int e^{-bx} \, dx.
\]

### 3. **Integrate \( \int e^{-bx} \, dx \):**

To integrate \( e^{-bx} \), use the following standard result:

\[
\int e^{-bx} \, dx = -\frac{1}{b} e^{-bx} + C,
\]

where \( C \) is the constant of integration.

### 4. **Combine the results:**

Multiply this result by the constants \( a \) and \( e^{-bt} \):

\[
\int a \cdot e^{-b(x + t)} \, dx = a \cdot e^{-bt} \left( -\frac{1}{b} e^{-bx} \right) + C.
\]

### 5. **Final simplified form:**

\[
\int a \cdot e^{-b(x + t)} \, dx = -\frac{a}{b} e^{-b(x + t)} + C.
\]

### Final Answer:

\[
\int a \cdot e^{-b(x + t)} \, dx = -\frac{a}{b} e^{-b(x + t)} + C.
\]

Would you like more details, or do you have any questions about this solution?

### Related Questions

1. How do you handle integrals involving products of exponential functions and polynomials?
2. What is the integral of \( e^{-bx} \) with respect to a different variable, say \( t \)?
3. Can you explain the properties of exponential functions used in integration?
4. How do you integrate an expression like \( e^{-bx^2} \)?
5. What changes in the integration if \( t \) were a variable instead of a constant?

### Tip

When integrating exponential functions, remember to pay close attention to the coefficients of the variable in the exponent; they often influence the integration through multiplicative factors like \( \frac{1}{b} \).

Learn how to solve the integral \int a \cdot e^{-b(x + t)}dx with step-by-step guidance. This problem involves key calculus concepts, such as exponential functions and integration. Suitable for undergraduate calculus students.

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