Calculus Examples

$\frac{d y}{d t} = k \sqrt[7]{t}$

Step 1

Rewrite the equation.

$d y = k \sqrt[7]{t} d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int k \sqrt[7]{t} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int k \sqrt[7]{t} d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$y + C_{1} = k \int \sqrt[7]{t} d t$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{t}$ as $t^{\frac{1}{7}}$ .

$y + C_{1} = k \int t^{\frac{1}{7}} d t$

Step 2.3.3

By the Power Rule, the integral of $t^{\frac{1}{7}}$ with respect to $t$ is $\frac{7}{8} t^{\frac{8}{7}}$ .

$y + C_{1} = k (\frac{7}{8} t^{\frac{8}{7}} + C_{2})$

Step 2.3.4

Simplify the answer.

Tap for more steps...

Step 2.3.4.1

Rewrite $k (\frac{7}{8} t^{\frac{8}{7}} + C_{2})$ as $k \frac{7}{8} t^{\frac{8}{7}} + C_{2}$ .

$y + C_{1} = k \frac{7}{8} t^{\frac{8}{7}} + C_{2}$

Step 2.3.4.2

Simplify.

Tap for more steps...

Step 2.3.4.2.1

Combine $k$ and $\frac{7}{8}$ .

$y + C_{1} = \frac{k \cdot 7}{8} t^{\frac{8}{7}} + C_{2}$

Step 2.3.4.2.2

Move $7$ to the left of $k$ .

$y + C_{1} = \frac{7 \cdot k}{8} t^{\frac{8}{7}} + C_{2}$

Step 2.3.4.2.3

Combine $\frac{7 k}{8}$ and $t^{\frac{8}{7}}$ .

$y + C_{1} = \frac{7 k t^{\frac{8}{7}}}{8} + C_{2}$

$y + C_{1} = \frac{7}{8} k t^{\frac{8}{7}} + C_{2}$

Step 2.3.5

Reorder terms.

$y + C_{1} = \frac{7}{8} t^{\frac{8}{7}} k + C_{2}$

Step 2.4

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

$y = \frac{7}{8} t^{\frac{8}{7}} k + K$