Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $(1 + x^{2}) \frac{d y}{d x} = 1$ by $1 + x^{2}$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $(1 + x^{2}) \frac{d y}{d x} = 1$ by $1 + x^{2}$ .

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

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $1 + x^{2}$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.3.2

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C_{2}$ .

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

Step 2.4

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

$y = \arctan (x) + K$