Calculus Examples

$\int 1.2 e^{0.6 x + 2} d x$

Step 1

Since $1.2$ is constant with respect to $x$ , move $1.2$ out of the integral.

$1.2 \int e^{0.6 x + 2} d x$

Step 2

Let $u = 0.6 x + 2$ . Then $d u = 0.6 d x$ , so $\frac{1}{0.6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.6 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $0.6 x + 2$ .

$\frac{d}{d x} [0.6 x + 2]$

Step 2.1.2

By the Sum Rule, the derivative of $0.6 x + 2$ with respect to $x$ is $\frac{d}{d x} [0.6 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [0.6 x] + \frac{d}{d x} [2]$

Step 2.1.3

Evaluate $\frac{d}{d x} [0.6 x]$ .

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Step 2.1.3.1

Since $0.6$ is constant with respect to $x$ , the derivative of $0.6 x$ with respect to $x$ is $0.6 \frac{d}{d x} [x]$ .

$0.6 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.1.3.2

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

$0.6 \cdot 1 + \frac{d}{d x} [2]$

Step 2.1.3.3

Multiply $0.6$ by $1$ .

$0.6 + \frac{d}{d x} [2]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0.6 + 0$

Step 2.1.4.2

Add $0.6$ and $0$ .

$0.6$

Step 2.2

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

$1.2 \int e^{u} \frac{1}{0.6} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{0.6}$ .

$1.2 \int \frac{e^{u}}{0.6} d u$

Step 4

Since $\frac{1}{0.6}$ is constant with respect to $u$ , move $\frac{1}{0.6}$ out of the integral.

$1.2 (\frac{1}{0.6} \int e^{u} d u)$

Step 5

Combine $\frac{1}{0.6}$ and $1.2$ .

$\frac{1.2}{0.6} \int e^{u} d u$

Step 6

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

$\frac{1.2}{0.6} (e^{u} + C)$

Step 7

Rewrite $\frac{1.2}{0.6} (e^{u} + C)$ as $2 e^{u} + C$ .

$2 e^{u} + C$

Step 8

Replace all occurrences of $u$ with $0.6 x + 2$ .

$2 e^{0.6 x + 2} + C$