The problem consists of two parts:

5(a): Use the integration by parts method to evaluate $\int (2x \cdot e^x) \, dx$. Show all steps, including the checking step.

5(b): Evaluate the definite integral $\int_0^1 (2x \cdot e^x) \, dx$.

Let's solve 5(a) step-by-step using the integration by parts formula:

Integration by Parts Formula:

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

Step 1: Choose $u$ and $dv$

• Let $u = 2x$ (because its derivative simplifies).
• Let $dv = e^x \, dx$ (because the integral of $e^x$ is itself).

Step 2: Compute $du$ and $v$

• $du = 2 \, dx$
• $v = e^x$

Step 3: Apply the formula

$\int (2x \cdot e^x) \, dx = uv - \int v \, du$ Substitute $u = 2x$, $v = e^x$, and $du = 2 \, dx$: $\int (2x \cdot e^x) \, dx = (2x \cdot e^x) - \int (e^x \cdot 2) \, dx$

Step 4: Simplify the remaining integral

$\int (2x \cdot e^x) \, dx = 2x \cdot e^x - 2 \int e^x \, dx$ The integral of $e^x$ is simply $e^x$, so: $\int (2x \cdot e^x) \, dx = 2x \cdot e^x - 2e^x + C$

Final Answer for 5(a):

$\int (2x \cdot e^x) \, dx = 2x \cdot e^x - 2e^x + C$

Checking the Result:

Differentiate $2x \cdot e^x - 2e^x + C$ to verify: $\frac{d}{dx} \big( 2x \cdot e^x - 2e^x + C \big) = 2e^x + 2x \cdot e^x - 2e^x = 2x \cdot e^x$ The derivative matches the original integrand, so the solution is correct.

Solve 5(b): Definite Integral

Evaluate $\int_0^1 (2x \cdot e^x) \, dx$: Use the result from 5(a): $\int_0^1 (2x \cdot e^x) \, dx = \Big[ 2x \cdot e^x - 2e^x \Big]_0^1$

Step 1: Evaluate at $x = 1$

$2(1) \cdot e^1 - 2e^1 = 2e - 2e = 0$

Step 2: Evaluate at $x = 0$

$2(0) \cdot e^0 - 2e^0 = 0 - 2 = -2$

Subtract: $\int_0^1 (2x \cdot e^x) \, dx = (0) - (-2) = 2$

Final Answer for 5(b):

$\int_0^1 (2x \cdot e^x) \, dx = 2$

Summary:

• 5(a): $\int (2x \cdot e^x) \, dx = 2x \cdot e^x - 2e^x + C$
• 5(b): $\int_0^1 (2x \cdot e^x) \, dx = 2$

Let me know if you'd like further clarifications or a deeper dive into any step!

Related Questions:

1. What are the conditions for applying integration by parts?
2. How do you verify the result of an indefinite integral?
3. Can the integration by parts formula be used iteratively?
4. How does the choice of $u$ and $dv$ affect the simplicity of solving an integral?
5. What are other methods for solving $\int (2x \cdot e^x) \, dx$?

Tip: Always confirm the result of indefinite integrals by differentiation to ensure correctness!