Calculus Examples

$\frac{x}{e^{x}}$

Step 1

Write $\frac{x}{e^{x}}$ as a function.

$f (x) = \frac{x}{e^{x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{x}{e^{x}} d x$

Step 4

Simplify the expression.

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Step 4.1

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

$\int x {(e^{x})}^{- 1} d x$

Step 4.2

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

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Step 4.2.1

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

$\int x e^{x \cdot - 1} d x$

Step 4.2.2

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

$\int x e^{- 1 \cdot x} d x$

Step 4.2.3

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

$\int x e^{- x} d x$

Step 5

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

$x (- e^{- x}) - \int - e^{- x} d x$

Step 6

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

$x (- e^{- x}) - - \int e^{- x} d x$

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

$x (- e^{- x}) + 1 \int e^{- x} d x$

Step 7.2

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

$x (- e^{- x}) + \int e^{- x} d x$

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $- x$ .

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

Step 8.1.2

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 8.1.3

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 8.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 8.2

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

$x (- e^{- x}) + \int - e^{u} d u$

Step 9

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

$x (- e^{- x}) - \int e^{u} d u$

Step 10

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

$x (- e^{- x}) - (e^{u} + C)$

Step 11

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

$- x e^{- x} - e^{u} + C$

Step 12

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

$- x e^{- x} - e^{- x} + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{x}{e^{x}}$ .

$F (x) =$ $- x e^{- x} - e^{- x} + C$