Math / Calculus

QuestionQuestion Evaluate $\int x e^{5 x} d x$ . Choose $u=x$ and $d v=e^{5 x} d x$ .

Studdy Solution

STEP 1

1. We are using integration by parts to solve the integral.
2. The formula for integration by parts is $\int u \, dv = uv - \int v \, du$ .
3. We have chosen $u = x$ and $dv = e^{5x} \, dx$ .

STEP 2

1. Identify $du$ and $v$ from the chosen $u$ and $dv$ .
2. Apply the integration by parts formula.
3. Simplify and solve the resulting integral.

STEP 3

Differentiate $u$ to find $du$ :
$u = x$ $du = dx$
Integrate $dv$ to find $v$ :
$dv = e^{5x} \, dx$
To find $v$ , integrate $e^{5x}$ :
$v = \int e^{5x} \, dx = \frac{1}{5} e^{5x}$

STEP 4

Apply the integration by parts formula $\int u \, dv = uv - \int v \, du$ :
$\int x e^{5x} \, dx = uv - \int v \, du$
Substitute the expressions for $u$ , $v$ , and $du$ :
$\int x e^{5x} \, dx = x \cdot \frac{1}{5} e^{5x} - \int \frac{1}{5} e^{5x} \, dx$
$= \frac{x}{5} e^{5x} - \frac{1}{5} \int e^{5x} \, dx$

STEP 5

Solve the remaining integral $\int e^{5x} \, dx$ :
$\int e^{5x} \, dx = \frac{1}{5} e^{5x}$
Substitute back into the expression:
$\frac{x}{5} e^{5x} - \frac{1}{5} \cdot \frac{1}{5} e^{5x}$
$= \frac{x}{5} e^{5x} - \frac{1}{25} e^{5x}$
Combine the terms:
$= \left(\frac{x}{5} - \frac{1}{25}\right) e^{5x}$
$= \frac{5x - 1}{25} e^{5x}$
Don't forget the constant of integration $C$ :
$= \frac{5x - 1}{25} e^{5x} + C$
The evaluated integral is:
$\boxed{\frac{5x - 1}{25} e^{5x} + C}$

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