Calculus Examples

$\frac{d V}{d t} = 4 \cos (2 t)$

Step 1

Rewrite the equation.

$d V = 4 \cos (2 t) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d V = \int 4 \cos (2 t) d t$

Step 2.2

Apply the constant rule.

$V + C_{1} = \int 4 \cos (2 t) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$V + C_{1} = 4 \int \cos (2 t) d t$

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $2 t$ .

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

Step 2.3.2.1.2

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 2.3.2.1.3

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 2.3.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.2.2

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

$V + C_{1} = 4 \int \cos (u) \frac{1}{2} d u$

Step 2.3.3

Combine $\cos (u)$ and $\frac{1}{2}$ .

$V + C_{1} = 4 \int \frac{\cos (u)}{2} d u$

Step 2.3.4

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

$V + C_{1} = 4 (\frac{1}{2} \int \cos (u) d u)$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{2}$ and $4$ .

$V + C_{1} = \frac{4}{2} \int \cos (u) d u$

Step 2.3.5.2

Cancel the common factor of $4$ and $2$ .

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Step 2.3.5.2.1

Factor $2$ out of $4$ .

$V + C_{1} = \frac{2 \cdot 2}{2} \int \cos (u) d u$

Step 2.3.5.2.2

Cancel the common factors.

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Step 2.3.5.2.2.1

Factor $2$ out of $2$ .

$V + C_{1} = \frac{2 \cdot 2}{2 (1)} \int \cos (u) d u$

Step 2.3.5.2.2.2

Cancel the common factor.

$V + C_{1} = \frac{2 \cdot 2}{2 \cdot 1} \int \cos (u) d u$

Step 2.3.5.2.2.3

Rewrite the expression.

$V + C_{1} = \frac{2}{1} \int \cos (u) d u$

Step 2.3.5.2.2.4

Divide $2$ by $1$ .

$V + C_{1} = 2 \int \cos (u) d u$

Step 2.3.6

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$V + C_{1} = 2 (\sin (u) + C_{2})$

Step 2.3.7

Simplify.

$V + C_{1} = 2 \sin (u) + C_{2}$

Step 2.3.8

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

$V + C_{1} = 2 \sin (2 t) + C_{2}$

Step 2.4

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

$V = 2 \sin (2 t) + K$