Math / Calculus

QuestionQuestion Evaluate using integration by parts. $\int x e^{-4 x} d x$

Studdy Solution

STEP 1

1. We are given the integral $\int x e^{-4x} \, dx$ .
2. We will use the integration by parts formula: $\int u \, dv = uv - \int v \, du$ .

STEP 2

1. Identify $u$ and $dv$ .
2. Differentiate $u$ to find $du$ .
3. Integrate $dv$ to find $v$ .
4. Apply the integration by parts formula.
5. Simplify and solve the remaining integral.

STEP 3

Identify $u$ and $dv$ from the integral $\int x e^{-4x} \, dx$ .
Let: $u = x$ $dv = e^{-4x} \, dx$

STEP 4

Differentiate $u$ to find $du$ .
$du = \frac{d}{dx}(x) \, dx = 1 \, dx$

STEP 5

Integrate $dv$ to find $v$ .
$v = \int e^{-4x} \, dx$
To integrate, use the substitution method: Let $w = -4x$ , then $dw = -4 \, dx$ or $dx = -\frac{1}{4} \, dw$ .
$v = \int e^{w} \left(-\frac{1}{4}\right) \, dw = -\frac{1}{4} \int e^{w} \, dw = -\frac{1}{4} e^{w} + C$
Substitute back for $w$ : $v = -\frac{1}{4} e^{-4x}$

STEP 6

Apply the integration by parts formula $\int u \, dv = uv - \int v \, du$ .
$\int x e^{-4x} \, dx = uv - \int v \, du$
Substitute $u$ , $v$ , and $du$ :
$= x \left(-\frac{1}{4} e^{-4x}\right) - \int \left(-\frac{1}{4} e^{-4x}\right) \cdot 1 \, dx$
$= -\frac{1}{4} x e^{-4x} + \frac{1}{4} \int e^{-4x} \, dx$

STEP 7

Simplify and solve the remaining integral $\int e^{-4x} \, dx$ .
We already found that: $\int e^{-4x} \, dx = -\frac{1}{4} e^{-4x}$
Substitute this back into the equation:
$= -\frac{1}{4} x e^{-4x} + \frac{1}{4} \left(-\frac{1}{4} e^{-4x}\right)$
$= -\frac{1}{4} x e^{-4x} - \frac{1}{16} e^{-4x}$
$= -\frac{1}{4} e^{-4x} (x + \frac{1}{4}) + C$
Where $C$ is the constant of integration.
The evaluated integral is:
$\boxed{-\frac{1}{4} e^{-4x} (x + \frac{1}{4}) + C}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms