Calculus Examples

$f (x) = (π + e^{\cot (π x)}) \csc^{2} (π x)$

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.

$F (x) = \int (π + e^{\cot (π x)}) \csc^{2} (π x) d x$

Step 3

Let $u_{2} = \cot (π x)$ . Then $d u_{2} = - π \csc^{2} (π x) d x$ , so $- \frac{1}{π} d u_{2} = \csc^{2} (π x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.1

Let $u_{2} = \cot (π x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 3.1.1

Differentiate $\cot (π x)$ .

$\frac{d}{d x} [\cot (π x)]$

Step 3.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = π x$ .

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Step 3.1.2.1

To apply the Chain Rule, set $u_{1}$ as $π x$ .

$\frac{d}{d u_{1}} [\cot (u_{1})] \frac{d}{d x} [π x]$

Step 3.1.2.2

The derivative of $\cot (u_{1})$ with respect to $u_{1}$ is $- \csc^{2} (u_{1})$ .

$- \csc^{2} (u_{1}) \frac{d}{d x} [π x]$

Step 3.1.2.3

Replace all occurrences of $u_{1}$ with $π x$ .

$- \csc^{2} (π x) \frac{d}{d x} [π x]$

Step 3.1.3

Differentiate.

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Step 3.1.3.1

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

$- \csc^{2} (π x) (π \frac{d}{d x} [x])$

Step 3.1.3.2

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

$- \csc^{2} (π x) (π \cdot 1)$

Step 3.1.3.3

Simplify the expression.

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Step 3.1.3.3.1

Multiply $π$ by $1$ .

$- \csc^{2} (π x) π$

Step 3.1.3.3.2

Reorder the factors of $- \csc^{2} (π x) π$ .

$- π \csc^{2} (π x)$

Step 3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int (- π - e^{u_{2}}) \frac{- 1}{- π} d u_{2}$

Step 4

Dividing two negative values results in a positive value.

$\int (- π - e^{u_{2}}) \frac{1}{π} d u_{2}$

Step 5

Since $\frac{1}{π}$ is constant with respect to $u_{2}$ , move $\frac{1}{π}$ out of the integral.

$\frac{1}{π} \int - π - e^{u_{2}} d u_{2}$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{π} (\int - π d u_{2} + \int - e^{u_{2}} d u_{2})$

Step 7

Apply the constant rule.

$\frac{1}{π} (- π u_{2} + C + \int - e^{u_{2}} d u_{2})$

Step 8

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\frac{1}{π} (- π u_{2} + C - \int e^{u_{2}} d u_{2})$

Step 9

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$\frac{1}{π} (- π u_{2} + C - (e^{u_{2}} + C))$

Step 10

Simplify.

$\frac{1}{π} (- π u_{2} - e^{u_{2}}) + C$

Step 11

Replace all occurrences of $u_{2}$ with $\cot (π x)$ .

$\frac{1}{π} (- π \cot (π x) - e^{\cot (π x)}) + C$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{1}{π} (- π \cot (π x)) + \frac{1}{π} (- e^{\cot (π x)}) + C$

Step 12.2

Cancel the common factor of $π$ .

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Step 12.2.1

Factor $π$ out of $- π \cot (π x)$ .

$\frac{1}{π} (π (- \cot (π x))) + \frac{1}{π} (- e^{\cot (π x)}) + C$

Step 12.2.2

Cancel the common factor.

$\frac{1}{π} (π (- \cot (π x))) + \frac{1}{π} (- e^{\cot (π x)}) + C$

Step 12.2.3

Rewrite the expression.

$- \cot (π x) + \frac{1}{π} (- e^{\cot (π x)}) + C$

Step 12.3

Combine $\frac{1}{π}$ and $e^{\cot (π x)}$ .

$- \cot (π x) - \frac{e^{\cot (π x)}}{π} + C$

Step 13

Reorder terms.

$- \cot (π x) - \frac{1}{π} e^{\cot (π x)} + C$

Step 14

The answer is the antiderivative of the function $f (x) = (π + e^{\cot (π x)}) \csc^{2} (π x)$ .

$F (x) =$ $- \cot (π x) - \frac{1}{π} e^{\cot (π x)} + C$