Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify the expression.

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Step 2.3.1.1

Reorder $x^{2}$ and $1$ .

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

Step 2.3.1.2

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 $K$ .

$y = \arctan (x) + K$