atic/chunks/29825-94580ffc5e6c1687.js","81779","static/chunks/81779-a1a04924c020dac8.js","28478","static/chunks/app/dn/qna/%5BquestionId%5D/page-673c40cf2d44650c.js"],"default"] 2c:T410,To evaluate the integral \( \int x e^x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by: \[ \int u \, dv = uv - \int v \, du \] ### Step 1: Choose \( u \) and \( dv \) Let: - \( u = x \) (which means \( du = dx \)) - \( dv = e^x \, dx \) (which means \( v = e^x \)) ### Step 2: Apply the integration by parts formula Using the integration by parts formula, we have: \[ \int x e^x \, dx = uv - \int v \, du \] Substituting the values of \( u \), \( du \), \( v \), and \( dv \): \[ \int x e^x \, dx = x e^x - \int e^x \, dx \] ### Step 3: Evaluate the remaining integral Now we need to evaluate \( \int e^x \, dx \): \[ \int e^x \, dx = e^x \] ### Step 4: Substitute back into the equation Now substituting this back into our equation: \[ \int x e^x \, dx = x e^x - e^x + C \] ### Step 5: Factor out \( e^x \) We can factor out \( e^x \): \[ \int x e^x \, dx = (x - 1)e^x + C \] ### Final Answer Thus, the final answer is: \[ \int x e^x \, dx = (x - 1)e^x + C \] ---2d:T495,To evaluate the integral \( \int x e^x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:
\[
\int u \, dv = uv - \int v \, du
\]
### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = x \) (which means \( du = dx \))
- \( dv = e^x \, dx \) (which means \( v = e^x \))
### Step 2: Apply the integration by parts formula
Using the integration by parts formula, we have:
\[
\int x e^x \, dx = uv - \int v \, du
\]
Substituting the values of \( u \), \( du \), \( v \), and \( dv \):
\[
\int x e^x \, dx = x e^x - \int e^x \, dx
\]
### Step 3: Evaluate the remaining integral
Now we need to evaluate \( \int e^x \, dx \):
\[
\int e^x \, dx = e^x
\]
### Step 4: Substitute

Evaluate `int xe^x dx`

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