Question
Use integration by parts to evaluate the integral \( \int x e^{x} \, dx \).
Ask by Bond Park. in the United States
Nov 21,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral \( \int x e^{x} \, dx \) evaluates to \( e^x (x - 1) + C \).
Solution
To evaluate the integral \( \int x e^{x} \, dx \) using integration by parts, we start by identifying the parts of the formula:
\[
\int u \, dv = uv - \int v \, du
\]
We choose:
- \( u = x \) (which means \( du = dx \))
- \( dv = e^x \, dx \) (which means \( v = e^x \))
Now we can apply the integration by parts formula:
1. Compute \( uv \):
\[
uv = x e^x
\]
2. Compute \( \int v \, du \):
\[
\int v \, du = \int e^x \, dx = e^x
\]
Now we can substitute these into the integration by parts formula:
\[
\int x e^x \, dx = x e^x - \int e^x \, dx
\]
Substituting the integral we computed:
\[
\int x e^x \, dx = x e^x - e^x + C
\]
where \( C \) is the constant of integration.
Finally, we can factor out \( e^x \):
\[
\int x e^x \, dx = e^x (x - 1) + C
\]
Thus, the evaluated integral is:
\[
\boxed{e^x (x - 1) + C}
\]
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Simplify this solution The Deep Dive
To evaluate the integral \( \int x e^{x} \, dx \) using integration by parts, we start by identifying our \( u \) and \( dv \). Let's set: - \( u = x \) (which means \( du = dx \)) - \( dv = e^{x} \, dx \) (which means \( v = e^{x} \)) Now, applying the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] we get: \[ \int x e^{x} \, dx = x e^{x} - \int e^{x} \, dx \] Now, calculate the integral \( \int e^{x} \, dx \): \[ \int e^{x} \, dx = e^{x} + C \] Substituting this back into our equation: \[ \int x e^{x} \, dx = x e^{x} - e^{x} + C \] We can simplify this to: \[ \int x e^{x} \, dx = e^{x}(x - 1) + C \] And that's our result!
