The image you uploaded contains a mathematical problem involving the calculation of an integral. The problem is asking you to evaluate the following indefinite integral:

$\int \left(1 - x\right) e^{x} \, dx$

Solution:

To solve the integral $\int \left(1 - x\right) e^{x} \, dx$, we can use integration by parts. The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Step 1: Identify $u$ and $dv$

Let:

• $u = 1 - x$, hence $du = -dx$
• $dv = e^x \, dx$, hence $v = e^x$

Step 2: Apply the integration by parts formula

$\int \left(1 - x\right) e^{x} \, dx = (1 - x)e^x - \int e^x(-dx)$

Step 3: Simplify the expression

$\int \left(1 - x\right) e^{x} \, dx = (1 - x)e^x + \int e^x \, dx$

$\int e^x \, dx = e^x$

Thus:

$\int \left(1 - x\right) e^{x} \, dx = (1 - x)e^x + e^x + C$

$\int \left(1 - x\right) e^{x} \, dx = e^x(1 - x + 1) + C$

$\int \left(1 - x\right) e^{x} \, dx = e^x(2 - x) + C$

Where $C$ is the constant of integration.

Final Answer:

$\int \left(1 - x\right) e^{x} \, dx = e^x(2 - x) + C$

Would you like further details on any of these steps?

Relative Questions:

1. How do you apply integration by parts to more complex integrals?
2. What is the method for solving integrals involving products of polynomials and exponentials?
3. Can you explain the importance of the constant of integration in indefinite integrals?
4. How does the choice of $u$ and $dv$ affect the integration by parts process?
5. What are some common mistakes to avoid when performing integration by parts?

Tip: Always double-check your $u$ and $dv$ choices in integration by parts to ensure the integration simplifies the problem effectively.