Calculus Examples

$\int 3 e^{6 x} d x$

Step 1

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

$3 \int e^{6 x} d x$

Step 2

Let $u = 6 x$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 6 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 2.1.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x]$

Step 2.1.3

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

$6 \cdot 1$

Step 2.1.4

Multiply $6$ by $1$ .

$6$

Step 2.2

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

$3 \int e^{u} \frac{1}{6} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{6}$ .

$3 \int \frac{e^{u}}{6} d u$

Step 4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$3 (\frac{1}{6} \int e^{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{6}$ and $3$ .

$\frac{3}{6} \int e^{u} d u$

Step 5.2

Cancel the common factor of $3$ and $6$ .

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Step 5.2.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{6} \int e^{u} d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 1}{3 \cdot 2} \int e^{u} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 2} \int e^{u} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{1}{2} \int e^{u} d u$

Step 6

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

$\frac{1}{2} (e^{u} + C)$

Step 7

Simplify.

$\frac{1}{2} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $6 x$ .

$\frac{1}{2} e^{6 x} + C$