$$ \begin{array}{ll} ext { Q3: Evaluate (Use integration by parts) } \ ext { 1) } \int 3 x e^{x} d x= \ 2 ext { of } 4 & 2\end{array} $$

Question

$$ \begin{array}{ll}	ext { Q3: Evaluate (Use integration by parts) } \ 	ext { 1) } \int 3 x e^{x} d x= \ 2 	ext { of } 4 & 2\end{array} $$

UpStudy AI · Accepted Answer

To evaluate the integral $$ \int 3x e^{x} \, dx $$ using integration by parts, follow these steps:

**Integration by Parts Formula:**
$$
\int u \, dv = uv - \int v \, du
$$

**Step 1: Choose $$ u $$ and $$ dv $$**
$$
u = 3x \quad \Rightarrow \quad du = 3 \, dx
$$
$$
dv = e^{x} \, dx \quad \Rightarrow \quad v = e^{x}
$$

**Step 2: Apply the Integration by Parts Formula**
$$
\int 3x e^{x} \, dx = u \cdot v - \int v \, du
$$
$$
= 3x e^{x} - \int e^{x} \cdot 3 \, dx
$$
$$
= 3x e^{x} - 3 \int e^{x} \, dx
$$

**Step 3: Integrate $$ \int e^{x} \, dx $$**
$$
\int e^{x} \, dx = e^{x} + C
$$
$$
\Rightarrow 3x e^{x} - 3 e^{x} + C
$$

**Final Answer:**
$$
\int 3x e^{x} \, dx = 3 e^{x} (x - 1) + C
$$
where $$ C $$ is the constant of integration.
$$ \int 3x e^{x} \, dx = 3 e^{x} (x - 1) + C $$

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions