Calculus Examples

$\frac{d y}{d x} = c_{1} x^{2} + c_{2} e^{2 x}$

Step 1

Rewrite the equation.

$d y = (c_{1} x^{2} + c_{2} e^{2 x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $c_{1}$ is constant with respect to $x$ , move $c_{1}$ out of the integral.

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

Step 2.3.3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 2.3.4

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

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

Step 2.3.5

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

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Step 2.3.5.1

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

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Step 2.3.5.1.1

Differentiate $2 x$ .

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

Step 2.3.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.3.5.1.3

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

$2 \cdot 1$

Step 2.3.5.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.5.2

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 2.3.6.2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 2.3.7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

Simplify.

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Step 2.3.10.1

Simplify.

$y + C_{1} = \frac{c_{1} x^{3}}{3} + \frac{c_{2}}{2} e^{u} + C_{4}$

Step 2.3.10.2

Combine $\frac{c_{2}}{2}$ and $e^{u}$ .

$y + C_{1} = \frac{c_{1} x^{3}}{3} + \frac{c_{2} e^{u}}{2} + C_{4}$

Step 2.3.11

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

$y + C_{1} = \frac{1}{3} c_{1} x^{3} + \frac{1}{2} c_{2} e^{2 x} + C_{4}$

Step 2.3.12

Reorder terms.

$y + C_{1} = \frac{1}{3} x^{3} c_{1} + \frac{1}{2} e^{2 x} c_{2} + C_{4}$

Step 2.4

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

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