Calculus Examples

$f (x) = \frac{1}{3^{x}}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \frac{1}{3^{x}} d x$

Step 3

Simplify the expression.

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Step 3.1

Negate the exponent of $3^{x}$ and move it out of the denominator.

$\int 1 {(3^{x})}^{- 1} d x$

Step 3.2

Simplify.

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Step 3.2.1

Multiply the exponents in ${(3^{x})}^{- 1}$ .

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Step 3.2.1.1

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

$\int 1 \cdot 3^{x \cdot - 1} d x$

Step 3.2.1.2

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

$\int 1 \cdot 3^{- 1 \cdot x} d x$

Step 3.2.1.3

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

$\int 1 \cdot 3^{- x} d x$

Step 3.2.2

Multiply $3^{- x}$ by $1$ .

$\int 3^{- x} d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $- x$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.2

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

$\int - 3^{u} d u$

Step 5

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

$- \int 3^{u} d u$

Step 6

The integral of $3^{u}$ with respect to $u$ is $\frac{3^{u}}{\ln (3)}$ .

$- (\frac{3^{u}}{\ln (3)} + C)$

Step 7

Rewrite $- (\frac{3^{u}}{\ln (3)} + C)$ as $- \frac{1}{\ln (3)} \cdot 3^{u} + C$ .

$- \frac{1}{\ln (3)} \cdot 3^{u} + C$

Step 8

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

$- \frac{1}{\ln (3)} \cdot 3^{- x} + C$

Step 9

The answer is the antiderivative of the function $f (x) = \frac{1}{3^{x}}$ .

$F (x) =$ $- \frac{1}{\ln (3)} \cdot 3^{- x} + C$