Calculus Examples

$x^{2} \sin (x) d x + x (y d) y = 0$

Step 1

Subtract $x^{2} \sin (x) d x$ from both sides of the equation.

$x y d y = - x^{2} \sin (x) d x$

Step 2

Multiply both sides by $\frac{1}{x}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Factor $x$ out of $x y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.3.2

Factor $x$ out of $x^{2} \sin (x)$ .

$y d y = \frac{- 1}{x} (x (x \sin (x))) d x$

Step 3.3.3

Cancel the common factor.

$y d y = \frac{- 1}{x} (x (x \sin (x))) d x$

Step 3.3.4

Rewrite the expression.

$y d y = - (x \sin (x)) d x$

$y d y = - x \sin (x) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int y d y = \int - x \sin (x) d x$

Step 4.2

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

$\frac{1}{2} y^{2} + C_{1} = \int - x \sin (x) d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - \int x \sin (x) d x$

Step 4.3.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (x)$ .

$\frac{1}{2} y^{2} + C_{1} = - (x (- \cos (x)) - \int - \cos (x) d x)$

Step 4.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = - (x (- \cos (x)) - - \int \cos (x) d x)$

Step 4.3.4

Simplify.

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Step 4.3.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} y^{2} + C_{1} = - (x (- \cos (x)) + 1 \int \cos (x) d x)$

Step 4.3.4.2

Multiply $\int \cos (x) d x$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = - (x (- \cos (x)) + \int \cos (x) d x)$

Step 4.3.5

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

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

Step 4.3.6

Rewrite $- (x (- \cos (x)) + \sin (x) + C_{2})$ as $- (- x \cos (x) + \sin (x)) + C_{2}$ .

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

Step 4.4

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

$\frac{1}{2} y^{2} = - (- x \cos (x) + \sin (x)) + K$

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- (- x \cos (x) + \sin (x)) + K)$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 5.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (- (- x \cos (x) + \sin (x)) + K)$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (- (- x \cos (x) + \sin (x)) + K)$

Step 5.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- (- x \cos (x) + \sin (x)) + K)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 (- (- x \cos (x) + \sin (x)) + K)$ .

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Step 5.2.2.1.1

Simplify each term.

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Step 5.2.2.1.1.1

Apply the distributive property.

$y^{2} = 2 (- (- x \cos (x)) - \sin (x) + K)$

Step 5.2.2.1.1.2

Multiply $- (- x \cos (x))$ .

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Step 5.2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$y^{2} = 2 (1 (x \cos (x)) - \sin (x) + K)$

Step 5.2.2.1.1.2.2

Multiply $x$ by $1$ .

$y^{2} = 2 (x \cos (x) - \sin (x) + K)$

Step 5.2.2.1.2

Apply the distributive property.

$y^{2} = 2 (x \cos (x)) + 2 (- \sin (x)) + 2 K$

Step 5.2.2.1.3

Multiply $- 1$ by $2$ .

$y^{2} = 2 x \cos (x) - 2 \sin (x) + 2 K$

Step 5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{2 x \cos (x) - 2 \sin (x) + 2 K}$

Step 5.4

Factor $2$ out of $2 x \cos (x) - 2 \sin (x) + 2 K$ .

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Step 5.4.1

Factor $2$ out of $2 x \cos (x)$ .

$y = \pm \sqrt{2 (x \cos (x)) - 2 \sin (x) + 2 K}$

Step 5.4.2

Factor $2$ out of $- 2 \sin (x)$ .

$y = \pm \sqrt{2 (x \cos (x)) + 2 (- \sin (x)) + 2 K}$

Step 5.4.3

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (x \cos (x)) + 2 (- \sin (x)) + 2 K}$

Step 5.4.4

Factor $2$ out of $2 (x \cos (x)) + 2 (- \sin (x))$ .

$y = \pm \sqrt{2 (x \cos (x) - \sin (x)) + 2 K}$

Step 5.4.5

Factor $2$ out of $2 (x \cos (x) - \sin (x)) + 2 K$ .

$y = \pm \sqrt{2 (x \cos (x) - \sin (x) + K)}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (x \cos (x) - \sin (x) + K)}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (x \cos (x) - \sin (x) + K)}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (x \cos (x) - \sin (x) + K)}$

$y = - \sqrt{2 (x \cos (x) - \sin (x) + K)}$

$y = \sqrt{2 (x \cos (x) - \sin (x) + K)}$

$y = - \sqrt{2 (x \cos (x) - \sin (x) + K)}$

$y = \sqrt{2 (x \cos (x) - \sin (x) + K)}$

$y = - \sqrt{2 (x \cos (x) - \sin (x) + K)}$