Calculus Examples

$\frac{d^{2} x}{d t^{2}} = e^{2 t}$

Step 1

Integrate both sides with respect to $t$ .

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Step 1.1

The first derivative is equal to the integral of the second derivative with respect to $t$ .

$\frac{d x}{d t} = \int e^{2 t} d t$

Step 1.2

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

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Step 1.2.1

Let $u_{1} = 2 t$ . Find $\frac{d u_{1}}{d t}$ .

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Step 1.2.1.1

Differentiate $2 t$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

$2 \cdot 1$

Step 1.2.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{d x}{d t} = \int e^{u_{1}} \frac{1}{2} d u_{1}$

Step 1.3

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

$\frac{d x}{d t} = \int \frac{e^{u_{1}}}{2} d u_{1}$

Step 1.4

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

$\frac{d x}{d t} = \frac{1}{2} \int e^{u_{1}} d u_{1}$

Step 1.5

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$\frac{d x}{d t} = \frac{1}{2} (e^{u_{1}} + C_{1})$

Step 1.6

Simplify.

$\frac{d x}{d t} = \frac{1}{2} e^{u_{1}} + C_{1}$

Step 1.7

Replace all occurrences of $u_{1}$ with $2 t$ .

$\frac{d x}{d t} = \frac{1}{2} e^{2 t} + C_{1}$

Step 2

Rewrite the equation.

$d x = (\frac{1}{2} e^{2 t} + C_{1}) d t$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int d x = \int \frac{1}{2} e^{2 t} + C_{1} d t$

Step 3.2

Apply the constant rule.

$x + C_{2} = \int \frac{1}{2} e^{2 t} + C_{1} d t$

Step 3.3

Integrate the right side.

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Step 3.3.1

Split the single integral into multiple integrals.

$x + C_{2} = \int \frac{1}{2} e^{2 t} d t + \int C_{1} d t$

Step 3.3.2

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

$x + C_{2} = \frac{1}{2} \int e^{2 t} d t + \int C_{1} d t$

Step 3.3.3

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

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Step 3.3.3.1

Let $u_{2} = 2 t$ . Find $\frac{d u_{2}}{d t}$ .

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Step 3.3.3.1.1

Differentiate $2 t$ .

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

Step 3.3.3.1.2

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

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

Step 3.3.3.1.3

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

$2 \cdot 1$

Step 3.3.3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.3.3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$x + C_{2} = \frac{1}{2} \int e^{u_{2}} \frac{1}{2} d u_{2} + \int C_{1} d t$

Step 3.3.4

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

$x + C_{2} = \frac{1}{2} \int \frac{e^{u_{2}}}{2} d u_{2} + \int C_{1} d t$

Step 3.3.5

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

$x + C_{2} = \frac{1}{2} (\frac{1}{2} \int e^{u_{2}} d u_{2}) + \int C_{1} d t$

Step 3.3.6

Simplify.

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Step 3.3.6.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$x + C_{2} = \frac{1}{2 \cdot 2} \int e^{u_{2}} d u_{2} + \int C_{1} d t$

Step 3.3.6.2

Multiply $2$ by $2$ .

$x + C_{2} = \frac{1}{4} \int e^{u_{2}} d u_{2} + \int C_{1} d t$

Step 3.3.7

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$x + C_{2} = \frac{1}{4} (e^{u_{2}} + C_{3}) + \int C_{1} d t$

Step 3.3.8

Apply the constant rule.

$x + C_{2} = \frac{1}{4} (e^{u_{2}} + C_{3}) + C_{1} t + C_{4}$

Step 3.3.9

Simplify.

$x + C_{2} = \frac{1}{4} e^{u_{2}} + C_{1} t + C_{5}$

Step 3.3.10

Replace all occurrences of $u_{2}$ with $2 t$ .

$x + C_{2} = \frac{1}{4} e^{2 t} + C_{1} t + C_{5}$

Step 3.4

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

$x = \frac{1}{4} e^{2 t} + C t + D$