Calculus Examples

$\frac{d N}{d t} = 400 + 600 \sqrt{t}$

Step 1

Rewrite the equation.

$d N = (400 + 600 \sqrt{t}) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d N = \int 400 + 600 \sqrt{t} d t$

Step 2.2

Apply the constant rule.

$N + C_{1} = \int 400 + 600 \sqrt{t} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$N + C_{1} = \int 400 d t + \int 600 \sqrt{t} d t$

Step 2.3.2

Apply the constant rule.

$N + C_{1} = 400 t + C_{2} + \int 600 \sqrt{t} d t$

Step 2.3.3

Since $600$ is constant with respect to $t$ , move $600$ out of the integral.

$N + C_{1} = 400 t + C_{2} + 600 \int \sqrt{t} d t$

Step 2.3.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$N + C_{1} = 400 t + C_{2} + 600 \int t^{\frac{1}{2}} d t$

Step 2.3.5

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

$N + C_{1} = 400 t + C_{2} + 600 (\frac{2}{3} t^{\frac{3}{2}} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$N + C_{1} = 400 t + 600 (\frac{2}{3}) t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $600$ and $\frac{2}{3}$ .

$N + C_{1} = 400 t + \frac{600 \cdot 2}{3} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.2

Multiply $600$ by $2$ .

$N + C_{1} = 400 t + \frac{1200}{3} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.3

Cancel the common factor of $1200$ and $3$ .

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Step 2.3.6.2.3.1

Factor $3$ out of $1200$ .

$N + C_{1} = 400 t + \frac{3 \cdot 400}{3} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.3.2

Cancel the common factors.

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Step 2.3.6.2.3.2.1

Factor $3$ out of $3$ .

$N + C_{1} = 400 t + \frac{3 \cdot 400}{3 (1)} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.3.2.2

Cancel the common factor.

$N + C_{1} = 400 t + \frac{3 \cdot 400}{3 \cdot 1} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.3.2.3

Rewrite the expression.

$N + C_{1} = 400 t + \frac{400}{1} t^{\frac{3}{2}} + C_{4}$

Step 2.3.6.2.3.2.4

Divide $400$ by $1$ .

$N + C_{1} = 400 t + 400 t^{\frac{3}{2}} + C_{4}$

Step 2.4

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

$N = 400 t + 400 t^{\frac{3}{2}} + K$