Calculus Examples

$\int \frac{l}{n π} \cdot \cos (\frac{n π x}{l}) d x$

Step 1

Combine $\frac{l}{n π}$ and $\cos (\frac{n π x}{l})$ .

$\int \frac{l \cos (\frac{n π x}{l})}{n π} d x$

Step 2

Since $\frac{l}{n π}$ is constant with respect to $x$ , move $\frac{l}{n π}$ out of the integral.

$\frac{l}{n π} \int \cos (\frac{n π x}{l}) d x$

Step 3

Let $u = \frac{n π x}{l}$ . Then $d u = \frac{π n}{l} d x$ , so $\frac{l}{π n} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \frac{n π x}{l}$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $\frac{n π x}{l}$ .

$\frac{d}{d x} [\frac{n π x}{l}]$

Step 3.1.2

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

$\frac{n π}{l} \frac{d}{d x} [x]$

Step 3.1.3

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

$\frac{n π}{l} \cdot 1$

Step 3.1.4

Simplify the expression.

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Step 3.1.4.1

Multiply $\frac{n π}{l}$ by $1$ .

$\frac{n π}{l}$

Step 3.1.4.2

Reorder terms.

$\frac{π n}{l}$

Step 3.2

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

$\frac{l}{n π} \int \cos (u) \frac{1}{\frac{π n}{l}} d u$

Step 4

Simplify.

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Step 4.1

Multiply by the reciprocal of the fraction to divide by $\frac{π n}{l}$ .

$\frac{l}{n π} \int \cos (u) (1 \frac{l}{π n}) d u$

Step 4.2

Multiply $\frac{l}{π n}$ by $1$ .

$\frac{l}{n π} \int \cos (u) \frac{l}{π n} d u$

Step 4.3

Combine $\cos (u)$ and $\frac{l}{π n}$ .

$\frac{l}{n π} \int \frac{\cos (u) l}{π n} d u$

Step 5

Since $\frac{l}{π n}$ is constant with respect to $u$ , move $\frac{l}{π n}$ out of the integral.

$\frac{l}{n π} (\frac{l}{π n} \int \cos (u) d u)$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{l}{π n}$ by $\frac{l}{n π}$ .

$\frac{l \cdot l}{π n (n π)} \int \cos (u) d u$

Step 6.2

Raise $l$ to the power of $1$ .

$\frac{l^{1} l}{π n (n π)} \int \cos (u) d u$

Step 6.3

Raise $l$ to the power of $1$ .

$\frac{l^{1} l^{1}}{π n (n π)} \int \cos (u) d u$

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{l^{1 + 1}}{π n (n π)} \int \cos (u) d u$

Step 6.5

Add $1$ and $1$ .

$\frac{l^{2}}{π n (n π)} \int \cos (u) d u$

Step 6.6

Raise $π$ to the power of $1$ .

$\frac{l^{2}}{n (n (π^{1} π))} \int \cos (u) d u$

Step 6.7

Raise $π$ to the power of $1$ .

$\frac{l^{2}}{n (n (π^{1} π^{1}))} \int \cos (u) d u$

Step 6.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{l^{2}}{n (n π^{1 + 1})} \int \cos (u) d u$

Step 6.9

Add $1$ and $1$ .

$\frac{l^{2}}{n (n π^{2})} \int \cos (u) d u$

Step 6.10

Raise $n$ to the power of $1$ .

$\frac{l^{2}}{n^{1} n π^{2}} \int \cos (u) d u$

Step 6.11

Raise $n$ to the power of $1$ .

$\frac{l^{2}}{n^{1} n^{1} π^{2}} \int \cos (u) d u$

Step 6.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{l^{2}}{n^{1 + 1} π^{2}} \int \cos (u) d u$

Step 6.13

Add $1$ and $1$ .

$\frac{l^{2}}{n^{2} π^{2}} \int \cos (u) d u$

Step 7

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{l^{2}}{n^{2} π^{2}} (\sin (u) + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{l^{2}}{n^{2} π^{2}} \sin (u) + C$

Step 8.2

Combine $\frac{l^{2}}{n^{2} π^{2}}$ and $\sin (u)$ .

$\frac{l^{2} \sin (u)}{n^{2} π^{2}} + C$

Step 9

Replace all occurrences of $u$ with $\frac{n π x}{l}$ .

$\frac{l^{2} \sin (\frac{n π x}{l})}{n^{2} π^{2}} + C$