Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $e^{x} \frac{d y}{d x} = 2 x$ by $e^{x}$ and simplify.

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Step 1.1.1

Divide each term in $e^{x} \frac{d y}{d x} = 2 x$ by $e^{x}$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $e^{x}$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Negate the exponent of $e^{x}$ and move it out of the denominator.

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

Step 2.3.2.2

Multiply the exponents in ${(e^{x})}^{- 1}$ .

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Step 2.3.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.2.2.2

Move $- 1$ to the left of $x$ .

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

Step 2.3.2.2.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.5.2

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

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

Step 2.3.6

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

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Step 2.3.6.1

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

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Step 2.3.6.1.1

Differentiate $- x$ .

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

Step 2.3.6.1.2

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 2.3.6.1.3

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 2.3.6.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.6.2

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.4

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

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