Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $x^{2} y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = y$ .

$x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x^{2} y' + y \frac{d}{d x} [x^{2}]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} y' + y (2 x)$

Step 1.4

Substitute $\frac{d y}{d x}$ for $y'$ .

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

Step 1.5

Remove parentheses.

$x^{2} \frac{d y}{d x} + y \cdot 2 x$

Step 1.6

Move $y$ .

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

Step 2

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x^{2} y] = \cos^{2} (x)$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x^{2} y] d x = \int \cos^{2} (x) d x$

Step 4

Integrate the left side.

$x^{2} y = \int \cos^{2} (x) d x$

Step 5

Integrate the right side.

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Step 5.1

Use the half-angle formula to rewrite $\cos^{2} (x)$ as $\frac{1 + \cos (2 x)}{2}$ .

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

Step 5.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 5.3

Split the single integral into multiple integrals.

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

Step 5.4

Apply the constant rule.

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

Step 5.5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 5.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 5.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 5.5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 5.6

Combine $\cos (u)$ and $\frac{1}{2}$ .

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

Step 5.7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 5.8

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

$x^{2} y = \frac{1}{2} (x + C + \frac{1}{2} (\sin (u) + C))$

Step 5.9

Simplify.

$x^{2} y = \frac{1}{2} (x + \frac{1}{2} \sin (u)) + C$

Step 5.10

Replace all occurrences of $u$ with $2 x$ .

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

Step 5.11

Simplify.

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Step 5.11.1

Combine $\frac{1}{2}$ and $\sin (2 x)$ .

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

Step 5.11.2

Apply the distributive property.

$x^{2} y = \frac{1}{2} x + \frac{1}{2} \cdot \frac{\sin (2 x)}{2} + C$

Step 5.11.3

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

$x^{2} y = \frac{x}{2} + \frac{1}{2} \cdot \frac{\sin (2 x)}{2} + C$

Step 5.11.4

Multiply $\frac{1}{2} \cdot \frac{\sin (2 x)}{2}$ .

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Step 5.11.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 x)}{2}$ .

$x^{2} y = \frac{x}{2} + \frac{\sin (2 x)}{2 \cdot 2} + C$

Step 5.11.4.2

Multiply $2$ by $2$ .

$x^{2} y = \frac{x}{2} + \frac{\sin (2 x)}{4} + C$

Step 5.12

Reorder terms.

$x^{2} y = \frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$

Step 6

Divide each term in $x^{2} y = \frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$ by $x^{2}$ and simplify.

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Step 6.1

Divide each term in $x^{2} y = \frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{2} x}{x^{2}} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{\frac{1}{2} x}{x^{2}} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2} x}{x^{2}} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

Factor $x$ out of $\frac{1}{2} x$ .

$y = \frac{x \frac{1}{2}}{x^{2}} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.1.2

Cancel the common factors.

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Step 6.3.1.1.2.1

Factor $x$ out of $x^{2}$ .

$y = \frac{x \frac{1}{2}}{x \cdot x} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.1.2.2

Cancel the common factor.

$y = \frac{x \frac{1}{2}}{x \cdot x} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{2}}{x} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{2} \cdot \frac{1}{x} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.3

Multiply $\frac{1}{2}$ by $\frac{1}{x}$ .

$y = \frac{1}{2 x} + \frac{\frac{1}{4} \sin (2 x)}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.4

Combine $\frac{1}{4}$ and $\sin (2 x)$ .

$y = \frac{1}{2 x} + \frac{\frac{\sin (2 x)}{4}}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.5

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{2 x} + \frac{\sin (2 x)}{4} \cdot \frac{1}{x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.6

Combine.

$y = \frac{1}{2 x} + \frac{\sin (2 x) \cdot 1}{4 x^{2}} + \frac{C}{x^{2}}$

Step 6.3.1.7

Multiply $\sin (2 x)$ by $1$ .

$y = \frac{1}{2 x} + \frac{\sin (2 x)}{4 x^{2}} + \frac{C}{x^{2}}$