Calculus Examples

$\frac{d y}{d t} = k \sqrt[5]{t}$

Step 1

Rewrite the equation.

$d y = k \sqrt[5]{t} d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int k \sqrt[5]{t} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int k \sqrt[5]{t} d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$y + C_{1} = k \int \sqrt[5]{t} d t$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{t}$ as $t^{\frac{1}{5}}$ .

$y + C_{1} = k \int t^{\frac{1}{5}} d t$

Step 2.3.3

By the Power Rule, the integral of $t^{\frac{1}{5}}$ with respect to $t$ is $\frac{5}{6} t^{\frac{6}{5}}$ .

$y + C_{1} = k (\frac{5}{6} t^{\frac{6}{5}} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $k (\frac{5}{6} t^{\frac{6}{5}} + C_{2})$ as $k \frac{5}{6} t^{\frac{6}{5}} + C_{2}$ .

$y + C_{1} = k \frac{5}{6} t^{\frac{6}{5}} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Combine $k$ and $\frac{5}{6}$ .

$y + C_{1} = \frac{k \cdot 5}{6} t^{\frac{6}{5}} + C_{2}$

Step 2.3.4.2.2

Move $5$ to the left of $k$ .

$y + C_{1} = \frac{5 \cdot k}{6} t^{\frac{6}{5}} + C_{2}$

Step 2.3.4.2.3

Combine $\frac{5 k}{6}$ and $t^{\frac{6}{5}}$ .

$y + C_{1} = \frac{5 k t^{\frac{6}{5}}}{6} + C_{2}$

$y + C_{1} = \frac{5}{6} k t^{\frac{6}{5}} + C_{2}$

Step 2.3.5

Reorder terms.

$y + C_{1} = \frac{5}{6} t^{\frac{6}{5}} k + C_{2}$

Step 2.4

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

$y = \frac{5}{6} t^{\frac{6}{5}} k + K$