Calculus Examples

$d y = 2 e^{3 x} d x$

Step 1

Set up an integral on each side.

$\int d y = \int 2 e^{3 x} d x$

Step 2

Apply the constant rule.

$y + C_{1} = \int 2 e^{3 x} d x$

Step 3

Integrate the right side.

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Step 3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 2 \int e^{3 x} d x$

Step 3.2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 3.2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 3.2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 3.2.1.3

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

$3 \cdot 1$

Step 3.2.1.4

Multiply $3$ by $1$ .

$3$

Step 3.2.2

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

$y + C_{1} = 2 \int e^{u} \frac{1}{3} d u$

Step 3.3

Combine $e^{u}$ and $\frac{1}{3}$ .

$y + C_{1} = 2 \int \frac{e^{u}}{3} d u$

Step 3.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$y + C_{1} = 2 (\frac{1}{3} \int e^{u} d u)$

Step 3.5

Combine $\frac{1}{3}$ and $2$ .

$y + C_{1} = \frac{2}{3} \int e^{u} d u$

Step 3.6

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

$y + C_{1} = \frac{2}{3} (e^{u} + C_{2})$

Step 3.7

Simplify.

$y + C_{1} = \frac{2}{3} e^{u} + C_{2}$

Step 3.8

Replace all occurrences of $u$ with $3 x$ .

$y + C_{1} = \frac{2}{3} e^{3 x} + C_{2}$

Step 4

Group the constant of integration on the right side as $K$ .

$y = \frac{2}{3} e^{3 x} + K$