Calculus Examples

$d y = 2 x d x$

Step 1

Set up an integral on each side.

$\int d y = \int 2 x d x$

Step 2

Apply the constant rule.

$y + C_{1} = \int 2 x d x$

Step 3

Integrate the right side.

Tap for more steps...

Step 3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 2 \int x d x$

Step 3.2

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

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

Step 3.3

Simplify the answer.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Combine $2$ and $\frac{1}{2}$ .

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

Step 3.3.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.2.1

Cancel the common factor.

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

Step 3.3.2.2.2

Rewrite the expression.

$y + C_{1} = 1 x^{2} + C_{2}$

Step 3.3.2.3

Multiply $x^{2}$ by $1$ .

$y + C_{1} = x^{2} + C_{2}$

Step 4

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

$y = x^{2} + K$