Calculus Examples

$\frac{d y}{d x} = x e^{x}$

Step 1

Rewrite the equation.

$d y = x e^{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x e^{x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x e^{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

$y + C_{1} = x e^{x} - \int e^{x} d x$

Step 2.3.2

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$y + C_{1} = x e^{x} - (e^{x} + C_{2})$

Step 2.3.3

Simplify.

$y + C_{1} = x e^{x} - e^{x} + C_{2}$

Step 2.3.4

Reorder terms.

$y + C_{1} = e^{x} x - e^{x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = e^{x} x - e^{x} + K$