Calculus Examples

$\int 9 e^{0.2 x} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int e^{0.2 x} d x$

Step 2

Let $u = 0.2 x$ . Then $d u = 0.2 d x$ , so $\frac{1}{0.2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $0.2 x$ .

$\frac{d}{d x} [0.2 x]$

Step 2.1.2

Since $0.2$ is constant with respect to $x$ , the derivative of $0.2 x$ with respect to $x$ is $0.2 \frac{d}{d x} [x]$ .

$0.2 \frac{d}{d x} [x]$

Step 2.1.3

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

$0.2 \cdot 1$

Step 2.1.4

Multiply $0.2$ by $1$ .

$0.2$

Step 2.2

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

$9 \int e^{u} \frac{1}{0.2} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{0.2}$ .

$9 \int \frac{e^{u}}{0.2} d u$

Step 4

Since $\frac{1}{0.2}$ is constant with respect to $u$ , move $\frac{1}{0.2}$ out of the integral.

$9 (\frac{1}{0.2} \int e^{u} d u)$

Step 5

Combine $\frac{1}{0.2}$ and $9$ .

$\frac{9}{0.2} \int e^{u} d u$

Step 6

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

$\frac{9}{0.2} (e^{u} + C)$

Step 7

Rewrite $\frac{9}{0.2} (e^{u} + C)$ as $45 e^{u} + C$ .

$45 e^{u} + C$

Step 8

Replace all occurrences of $u$ with $0.2 x$ .

$45 e^{0.2 x} + C$