Calculus Examples

$\frac{d y}{d x} = 10 x^{5}$

Step 1

Rewrite the equation.

$d y = 10 x^{5} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 10 x^{5} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 10 x^{5} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$y + C_{1} = 10 \int x^{5} d x$

Step 2.3.2

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

$y + C_{1} = 10 (\frac{1}{6} x^{6} + C_{2})$

Step 2.3.3

Simplify the answer.

Tap for more steps...

Step 2.3.3.1

Rewrite $10 (\frac{1}{6} x^{6} + C_{2})$ as $10 (\frac{1}{6}) x^{6} + C_{2}$ .

$y + C_{1} = 10 (\frac{1}{6}) x^{6} + C_{2}$

Step 2.3.3.2

Simplify.

Tap for more steps...

Step 2.3.3.2.1

Combine $10$ and $\frac{1}{6}$ .

$y + C_{1} = \frac{10}{6} x^{6} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $10$ and $6$ .

Tap for more steps...

Step 2.3.3.2.2.1

Factor $2$ out of $10$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.3.2.2.2.1

Factor $2$ out of $6$ .

$y + C_{1} = \frac{2 \cdot 5}{2 \cdot 3} x^{6} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$y + C_{1} = \frac{2 \cdot 5}{2 \cdot 3} x^{6} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$y + C_{1} = \frac{5}{3} x^{6} + C_{2}$

Step 2.4

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

$y = \frac{5}{3} x^{6} + K$