Calculus Examples

$\int \frac{1}{e^{4 x}} d x$

Step 1

Simplify the expression.

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Step 1.1

Negate the exponent of $e^{4 x}$ and move it out of the denominator.

$\int 1 {(e^{4 x})}^{- 1} d x$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the exponents in ${(e^{4 x})}^{- 1}$ .

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Step 1.2.1.1

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

$\int 1 e^{4 x \cdot - 1} d x$

Step 1.2.1.2

Multiply $- 1$ by $4$ .

$\int 1 e^{- 4 x} d x$

Step 1.2.2

Multiply $e^{- 4 x}$ by $1$ .

$\int e^{- 4 x} d x$

Step 2

Let $u = - 4 x$ . Then $d u = - 4 d x$ , so $- \frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = - 4 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 2.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 2.1.3

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

$- 4 \cdot 1$

Step 2.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 2.2

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

$\int e^{u} \frac{1}{- 4} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$\int e^{u} (- \frac{1}{4}) d u$

Step 3.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$\int - \frac{e^{u}}{4} d u$

Step 4

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

$- \int \frac{e^{u}}{4} d u$

Step 5

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- (\frac{1}{4} \int e^{u} d u)$

Step 6

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

$- \frac{1}{4} (e^{u} + C)$

Step 7

Simplify.

$- \frac{1}{4} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $- 4 x$ .

$- \frac{1}{4} e^{- 4 x} + C$