Calculus Examples

$2 \frac{d y}{d x} = 4 x e^{- x}$

Step 1

Rewrite the equation.

$2 d y = 4 x e^{- x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 d y = \int 4 x e^{- x} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.2

Multiply $\int e^{- x} d x$ by $1$ .

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

Step 2.3.5

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.5.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.5.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 2.3.5.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 2.3.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 2.3.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

Rewrite $4 (x (- e^{- x}) - (e^{u} + C_{2}))$ as $4 (- x e^{- x} - e^{u}) + C_{2}$ .

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

Step 2.3.9

Replace all occurrences of $u$ with $- x$ .

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

Step 2.4

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

$2 y = 4 (- x e^{- x} - e^{- x}) + K$

Step 3

Divide each term in $2 y = 4 (- x e^{- x} - e^{- x}) + K$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 y = 4 (- x e^{- x} - e^{- x}) + K$ by $2$ .

$\frac{2 y}{2} = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 (- x e^{- x} - e^{- x})}{2} + \frac{K}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$y = \frac{4 (- x e^{- x} - e^{- x}) + K}{2}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Apply the distributive property.

$y = \frac{4 (- x e^{- x}) + 4 (- e^{- x}) + K}{2}$

Step 3.3.2.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 (x e^{- x}) + 4 (- e^{- x}) + K}{2}$

Step 3.3.2.3

Multiply $- 1$ by $4$ .

$y = \frac{- 4 x e^{- x} - 4 e^{- x} + K}{2}$

Step 3.3.3

Simplify with factoring out.

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Step 3.3.3.1

Factor $- 1$ out of $- 4 x e^{- x}$ .

$y = \frac{- (4 x e^{- x}) - 4 e^{- x} + K}{2}$

Step 3.3.3.2

Factor $- 1$ out of $- 4 e^{- x}$ .

$y = \frac{- (4 x e^{- x}) - (4 e^{- x}) + K}{2}$

Step 3.3.3.3

Factor $- 1$ out of $- (4 x e^{- x}) - (4 e^{- x})$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x}) + K}{2}$

Step 3.3.3.4

Factor $- 1$ out of $K$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x}) - 1 (- K)}{2}$

Step 3.3.3.5

Factor $- 1$ out of $- (4 x e^{- x} + 4 e^{- x}) - 1 (- K)$ .

$y = \frac{- (4 x e^{- x} + 4 e^{- x} - K)}{2}$

Step 3.3.3.6

Simplify the expression.

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Step 3.3.3.6.1

Rewrite $- (4 x e^{- x} + 4 e^{- x} - K)$ as $- 1 (4 x e^{- x} + 4 e^{- x} - K)$ .

$y = \frac{- 1 (4 x e^{- x} + 4 e^{- x} - K)}{2}$

Step 3.3.3.6.2

Move the negative in front of the fraction.

$y = - \frac{4 x e^{- x} + 4 e^{- x} - K}{2}$

Step 4

Simplify the constant of integration.

$y = - \frac{4 x e^{- x} + 4 e^{- x} + K}{2}$