Let's evaluate the given integral using integration by parts.

The expression you're working with is $\int 6t e^t \, dt$. Since no bounds are provided, we'll solve the indefinite integral.

Integration by Parts Formula

The formula for integration by parts is:

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

Step 1: Choose $u$ and $dv$

To apply integration by parts, we first need to identify $u$ and $dv$. In general, we choose $u$ to be the part that simplifies when differentiated, and $dv$ to be the part that can be easily integrated.

Let:

• $u = 6t$ (which simplifies upon differentiation),
• $dv = e^t \, dt$ (which is easily integrated).

Step 2: Differentiate $u$ and Integrate $dv$

Now, we differentiate $u$ and integrate $dv$:

• $du = 6 \, dt$,
• $v = e^t$.

Step 3: Apply the Integration by Parts Formula

Using the formula $\int u \, dv = uv - \int v \, du$, we can now substitute the values:

$\int 6t e^t \, dt = 6t \cdot e^t - \int e^t \cdot 6 \, dt$

Step 4: Simplify the Remaining Integral

The remaining integral is:

$\int 6 e^t \, dt$

The integral of $e^t$ is simply $e^t$, so:

$\int 6 e^t \, dt = 6 e^t$

Step 5: Final Expression

Substitute this result back into the equation:

$\int 6t e^t \, dt = 6t e^t - 6 e^t + C$

Thus, the indefinite integral of $6t e^t$ is:

$\int 6t e^t \, dt = 6 e^t (t - 1) + C$

Where $C$ is the constant of integration.

Would you like me to clarify any steps or perform additional operations, such as evaluating a definite integral if bounds are given?

Related Questions:

1. How do you evaluate definite integrals using integration by parts?
2. Can integration by parts be applied to trigonometric functions?
3. What is the intuition behind the choice of $u$ and $dv$ in integration by parts?
4. How would this problem change if the exponential term were more complex (e.g., $e^{2t}$)?
5. How is integration by parts related to the product rule for differentiation?

Tip:

When using integration by parts, always choose $u$ to be a function that simplifies when differentiated!