Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{1 + y^{2}}$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2

Combine $2$ and $\frac{1}{1 + y^{2}}$ .

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

Step 1.2.3

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

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Step 1.2.3.1

Factor $1 + y^{2}$ out of $(1 + y^{2}) x$ .

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

Step 1.2.3.2

Cancel the common factor.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

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

Step 2.3.3.2.2.2

Rewrite the expression.

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

Step 2.3.3.2.3

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

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

Step 2.4

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

$\arctan (y) = x^{2} + K$

Step 3

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

$y = \tan (x^{2} + K)$