Calculus Examples

$\frac{d y}{d t} = \frac{d}{d t} (x^{2} + 4)$

Step 1

Rewrite the equation.

$d y = \frac{d}{d t} (x^{2} + 4) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{d}{d t} (x^{2} + 4) d t$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply the constant rule.

$y + C_{1} = \frac{d}{d t} (x^{2} + 4) t + C_{2}$

Step 2.3.2

Simplify the answer.

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Step 2.3.2.1

Combine $t$ and $\frac{d}{d t}$ .

$y + C_{1} = \frac{t d}{d t} (x^{2} + 4) + C_{2}$

Step 2.3.2.2

Rewrite $\frac{t d}{d t} (x^{2} + 4) + C_{2}$ as $\frac{1}{d t} t d (x^{2} + 4) + C_{2}$ .

$y + C_{1} = \frac{1}{d t} t d (x^{2} + 4) + C_{2}$

Step 2.3.3

Reorder terms.

$y + C_{1} = \frac{1}{d t} d t (x^{2} + 4) + C_{2}$

Step 2.4

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

$y = \frac{1}{d t} d t (x^{2} + 4) + K$