Calculus Examples

$\frac{d y (t)}{d t} = 2 t^{2}$

Step 1

Separate the variables.

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Step 1.1

Factor $\frac{d y}{d t}$ out.

$t \frac{d y}{d t} = 2 t^{2}$

Step 1.2

Divide each term in $t \frac{d y}{d t} = 2 t^{2}$ by $t$ and simplify.

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Step 1.2.1

Divide each term in $t \frac{d y}{d t} = 2 t^{2}$ by $t$ .

$\frac{t \frac{d y}{d t}}{t} = \frac{2 t^{2}}{t}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $t$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{t \frac{d y}{d t}}{t} = \frac{2 t^{2}}{t}$

Step 1.2.2.1.2

Divide $\frac{d y}{d t}$ by $1$ .

$\frac{d y}{d t} = \frac{2 t^{2}}{t}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $t^{2}$ and $t$ .

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Step 1.2.3.1.1

Factor $t$ out of $2 t^{2}$ .

$\frac{d y}{d t} = \frac{t (2 t)}{t}$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Raise $t$ to the power of $1$ .

$\frac{d y}{d t} = \frac{t (2 t)}{t^{1}}$

Step 1.2.3.1.2.2

Factor $t$ out of $t^{1}$ .

$\frac{d y}{d t} = \frac{t (2 t)}{t \cdot 1}$

Step 1.2.3.1.2.3

Cancel the common factor.

$\frac{d y}{d t} = \frac{t (2 t)}{t \cdot 1}$

Step 1.2.3.1.2.4

Rewrite the expression.

$\frac{d y}{d t} = \frac{2 t}{1}$

Step 1.2.3.1.2.5

Divide $2 t$ by $1$ .

$\frac{d y}{d t} = 2 t$

Step 1.3

Rewrite the equation.

$d y = 2 t d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 2 t d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 2 t d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = 2 \int t d t$

Step 2.3.2

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

$y + C_{1} = 2 (\frac{1}{2} t^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{2} t^{2} + C_{2})$ as $2 (\frac{1}{2}) t^{2} + C_{2}$ .

$y + C_{1} = 2 (\frac{1}{2}) t^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$y + C_{1} = \frac{2}{2} t^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$y + C_{1} = \frac{2}{2} t^{2} + C_{2}$

Step 2.3.3.2.2.2

Rewrite the expression.

$y + C_{1} = 1 t^{2} + C_{2}$

Step 2.3.3.2.3

Multiply $t^{2}$ by $1$ .

$y + C_{1} = t^{2} + C_{2}$

Step 2.4

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

$y = t^{2} + K$