Calculus Examples

$\frac{1}{\cos^{2} (y)} d x - (6 x + 1) d y = 0$

Step 1

Subtract $\frac{1}{\cos^{2} (y)} d x$ from both sides of the equation.

$- (6 x + 1) d y = - \frac{1}{\cos^{2} (y)} d x$

Step 2

Multiply both sides by $\frac{\cos^{2} (y)}{6 x + 1}$ .

$\frac{\cos^{2} (y)}{6 x + 1} (- (6 x + 1)) d y = \frac{\cos^{2} (y)}{6 x + 1} (- \frac{1}{\cos^{2} (y)}) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$- \frac{\cos^{2} (y)}{6 x + 1} (6 x + 1) d y = \frac{\cos^{2} (y)}{6 x + 1} (- \frac{1}{\cos^{2} (y)}) d x$

Step 3.2

Cancel the common factor of $6 x + 1$ .

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Step 3.2.1

Move the leading negative in $- \frac{\cos^{2} (y)}{6 x + 1}$ into the numerator.

$\frac{- \cos^{2} (y)}{6 x + 1} (6 x + 1) d y = \frac{\cos^{2} (y)}{6 x + 1} (- \frac{1}{\cos^{2} (y)}) d x$

Step 3.2.2

Cancel the common factor.

$\frac{- \cos^{2} (y)}{6 x + 1} (6 x + 1) d y = \frac{\cos^{2} (y)}{6 x + 1} (- \frac{1}{\cos^{2} (y)}) d x$

Step 3.2.3

Rewrite the expression.

$- \cos^{2} (y) d y = \frac{\cos^{2} (y)}{6 x + 1} (- \frac{1}{\cos^{2} (y)}) d x$

Step 3.3

Rewrite using the commutative property of multiplication.

$- \cos^{2} (y) d y = - \frac{\cos^{2} (y)}{6 x + 1} \cdot \frac{1}{\cos^{2} (y)} d x$

Step 3.4

Cancel the common factor of $\cos^{2} (y)$ .

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Step 3.4.1

Move the leading negative in $- \frac{\cos^{2} (y)}{6 x + 1}$ into the numerator.

$- \cos^{2} (y) d y = \frac{- \cos^{2} (y)}{6 x + 1} \cdot \frac{1}{\cos^{2} (y)} d x$

Step 3.4.2

Factor $\cos^{2} (y)$ out of $- \cos^{2} (y)$ .

$- \cos^{2} (y) d y = \frac{\cos^{2} (y) \cdot - 1}{6 x + 1} \cdot \frac{1}{\cos^{2} (y)} d x$

Step 3.4.3

Cancel the common factor.

$- \cos^{2} (y) d y = \frac{\cos^{2} (y) \cdot - 1}{6 x + 1} \cdot \frac{1}{\cos^{2} (y)} d x$

Step 3.4.4

Rewrite the expression.

$- \cos^{2} (y) d y = \frac{- 1}{6 x + 1} d x$

Step 3.5

Move the negative in front of the fraction.

$- \cos^{2} (y) d y = - \frac{1}{6 x + 1} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int - \cos^{2} (y) d y = \int - \frac{1}{6 x + 1} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

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

$- \int \cos^{2} (y) d y = \int - \frac{1}{6 x + 1} d x$

Step 4.2.2

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

$- \int \frac{1 + \cos (2 y)}{2} d y = \int - \frac{1}{6 x + 1} d x$

Step 4.2.3

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

$- (\frac{1}{2} \int 1 + \cos (2 y) d y) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.4

Split the single integral into multiple integrals.

$- \frac{1}{2} (\int d y + \int \cos (2 y) d y) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.5

Apply the constant rule.

$- \frac{1}{2} (y + C_{1} + \int \cos (2 y) d y) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.6

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

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Step 4.2.6.1

Let $u_{1} = 2 y$ . Find $\frac{d u_{1}}{d y}$ .

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Step 4.2.6.1.1

Differentiate $2 y$ .

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

Step 4.2.6.1.2

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

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

Step 4.2.6.1.3

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

$2 \cdot 1$

Step 4.2.6.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2.6.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$- \frac{1}{2} (y + C_{1} + \int \cos (u_{1}) \frac{1}{2} d u_{1}) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.7

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

$- \frac{1}{2} (y + C_{1} + \int \frac{\cos (u_{1})}{2} d u_{1}) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.8

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

$- \frac{1}{2} (y + C_{1} + \frac{1}{2} \int \cos (u_{1}) d u_{1}) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.9

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$- \frac{1}{2} (y + C_{1} + \frac{1}{2} (\sin (u_{1}) + C_{2})) = \int - \frac{1}{6 x + 1} d x$

Step 4.2.10

Simplify.

$- \frac{1}{2} (y + \frac{1}{2} \sin (u_{1})) + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.11

Replace all occurrences of $u_{1}$ with $2 y$ .

$- \frac{1}{2} (y + \frac{1}{2} \sin (2 y)) + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.12

Simplify.

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Step 4.2.12.1

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

$- \frac{1}{2} (y + \frac{\sin (2 y)}{2}) + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.12.2

Apply the distributive property.

$- \frac{1}{2} y - \frac{1}{2} \cdot \frac{\sin (2 y)}{2} + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.12.3

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

$- \frac{y}{2} - \frac{1}{2} \cdot \frac{\sin (2 y)}{2} + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.12.4

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

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Step 4.2.12.4.1

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

$- \frac{y}{2} - \frac{\sin (2 y)}{2 \cdot 2} + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.12.4.2

Multiply $2$ by $2$ .

$- \frac{y}{2} - \frac{\sin (2 y)}{4} + C_{3} = \int - \frac{1}{6 x + 1} d x$

Step 4.2.13

Reorder terms.

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = \int - \frac{1}{6 x + 1} d x$

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.

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \int \frac{1}{6 x + 1} d x$

Step 4.3.2

Let $u_{2} = 6 x + 1$ . Then $d u_{2} = 6 d x$ , so $\frac{1}{6} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.3.2.1

Let $u_{2} = 6 x + 1$ . Find $\frac{d u_{2}}{d x}$ .

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Step 4.3.2.1.1

Differentiate $6 x + 1$ .

$\frac{d}{d x} [6 x + 1]$

Step 4.3.2.1.2

By the Sum Rule, the derivative of $6 x + 1$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 4.3.2.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 4.3.2.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 4.3.2.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [1]$

Step 4.3.2.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [1]$

Step 4.3.2.1.4

Differentiate using the Constant Rule.

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Step 4.3.2.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$6 + 0$

Step 4.3.2.1.4.2

Add $6$ and $0$ .

$6$

Step 4.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \int \frac{1}{u_{2}} \cdot \frac{1}{6} d u_{2}$

Step 4.3.3

Simplify.

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Step 4.3.3.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{6}$ .

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \int \frac{1}{u_{2} \cdot 6} d u_{2}$

Step 4.3.3.2

Move $6$ to the left of $u_{2}$ .

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \int \frac{1}{6 u_{2}} d u_{2}$

Step 4.3.4

Since $\frac{1}{6}$ is constant with respect to $u_{2}$ , move $\frac{1}{6}$ out of the integral.

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - (\frac{1}{6} \int \frac{1}{u_{2}} d u_{2})$

Step 4.3.5

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \frac{1}{6} (\ln (| u_{2} |) + C_{4})$

Step 4.3.6

Simplify.

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \frac{1}{6} \ln (| u_{2} |) + C_{4}$

Step 4.3.7

Replace all occurrences of $u_{2}$ with $6 x + 1$ .

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) + C_{3} = - \frac{1}{6} \ln (| 6 x + 1 |) + C_{4}$

Step 4.4

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

$- \frac{1}{2} y - \frac{1}{4} \sin (2 y) = - \frac{1}{6} \ln (| 6 x + 1 |) + C$