Calculus Examples

$\cos (e c x)$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $e c x$ .

$\frac{d}{d c} [e c x]$

Step 1.1.2

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

$e x \frac{d}{d c} [c]$

Step 1.1.3

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

$e x \cdot 1$

Step 1.1.4

Simplify the expression.

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Step 1.1.4.1

Multiply $e$ by $1$ .

$x \cdot e$

Step 1.1.4.2

Reorder the factors of $x e$ .

$e x$

Step 1.2

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

$\int \cos (u) \frac{1}{e x} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{e x}$ .

$\int \frac{\cos (u)}{e x} d u$

Step 3

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

$\frac{1}{e x} \int \cos (u) d u$

Step 4

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

$\frac{1}{e x} (\sin (u) + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$\frac{1}{e x} \sin (u) + C$

Step 5.2

Combine $\frac{1}{e x}$ and $\sin (u)$ .

$\frac{\sin (u)}{e x} + C$

Step 6

Replace all occurrences of $u$ with $e c x$ .

$\frac{\sin (e c x)}{e x} + C$