Calculus Examples

$\frac{1}{e^{- x}}$

Step 1

Write $\frac{1}{e^{- x}}$ as a function.

$f (x) = \frac{1}{e^{- x}}$

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.

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

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.

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

Step 4.2

Simplify.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply $- x \cdot - 1$ .

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Step 4.2.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.1.2.2

Multiply $x$ by $1$ .

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

Step 4.2.2

Multiply $e^{x}$ by $1$ .

$\int e^{x} d x$

Step 5

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

$e^{x} + C$

Step 6

The answer is the antiderivative of the function $f (x) = \frac{1}{e^{- x}}$ .

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