Calculus Examples

$\frac{d^{2} y}{d x^{2}} = x e^{x}$

Step 1

Integrate both sides with respect to $x$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $x$ .

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

Step 1.2

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

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

Step 1.3

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

$\frac{d y}{d x} = x e^{x} - (e^{x} + C_{1})$

Step 1.4

Simplify.

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

Step 2

Rewrite the equation.

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

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

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

Step 3.2

Apply the constant rule.

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

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

Step 3.3.2

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

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

Step 3.3.3

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

$y + C_{2} = x e^{x} - (e^{x} + C_{3}) + \int - e^{x} d x + \int C_{1} d x$

Step 3.3.4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{2} = x e^{x} - (e^{x} + C_{3}) - \int e^{x} d x + \int C_{1} d x$

Step 3.3.5

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

$y + C_{2} = x e^{x} - (e^{x} + C_{3}) - (e^{x} + C_{4}) + \int C_{1} d x$

Step 3.3.6

Apply the constant rule.

$y + C_{2} = x e^{x} - (e^{x} + C_{3}) - (e^{x} + C_{4}) + C_{1} x + C_{5}$

Step 3.3.7

Simplify.

$y + C_{2} = x e^{x} - 2 e^{x} + C_{1} x + C_{6}$

Step 3.3.8

Reorder terms.

$y + C_{2} = e^{x} x + C_{1} x - 2 e^{x} + C_{6}$

Step 3.4

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

$y = e^{x} x + C x - 2 e^{x} + D$