Calculus Examples

$f' (x) = \frac{1}{3} \cdot \csc^{2} (\frac{1}{3} x)$

Step 1

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

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

Step 2

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \csc^{2} (\frac{1}{3} x) d x$

Step 3

Let $u = \frac{1}{3} x$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \frac{1}{3} x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $\frac{1}{3} x$ .

$\frac{d}{d x} [\frac{1}{3} x]$

Step 3.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 3.1.3

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

$\frac{1}{3} \cdot 1$

Step 3.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 3.2

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

$\frac{1}{3} \int \csc^{2} (u) \frac{1}{\frac{1}{3}} d u$

Step 4

Simplify.

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Step 4.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$\frac{1}{3} \int \csc^{2} (u) (1 \cdot 3) d u$

Step 4.2

Multiply $3$ by $1$ .

$\frac{1}{3} \int \csc^{2} (u) \cdot 3 d u$

Step 4.3

Move $3$ to the left of $\csc^{2} (u)$ .

$\frac{1}{3} \int 3 \csc^{2} (u) d u$

Step 5

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$\frac{1}{3} (3 \int \csc^{2} (u) d u)$

Step 6

Simplify.

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Step 6.1

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} \int \csc^{2} (u) d u$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Cancel the common factor.

$\frac{3}{3} \int \csc^{2} (u) d u$

Step 6.2.2

Rewrite the expression.

$1 \int \csc^{2} (u) d u$

Step 6.3

Multiply $\int \csc^{2} (u) d u$ by $1$ .

$\int \csc^{2} (u) d u$

Step 7

Since the derivative of $- \cot (u)$ is $\csc^{2} (u)$ , the integral of $\csc^{2} (u)$ is $- \cot (u)$ .

$- \cot (u) + C$

Step 8

Replace all occurrences of $u$ with $\frac{1}{3} x$ .

$- \cot (\frac{1}{3} x) + C$

Step 9

The function $f$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$f (x) = - \cot (\frac{1}{3} x) + C$