Calculus Examples

$\frac{d y}{d x} = 4 y^{3} \cos^{2} (x)$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $\frac{1}{y^{3}}$ .

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (4 y^{3} \cos^{2} (x))$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{3}} \frac{d y}{d x} = 4 \frac{1}{y^{3}} (y^{3} \cos^{2} (x))$

Step 1.2.2

Combine $4$ and $\frac{1}{y^{3}}$ .

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{4}{y^{3}} (y^{3} \cos^{2} (x))$

Step 1.2.3

Cancel the common factor of $y^{3}$ .

Tap for more steps...

Step 1.2.3.1

Factor $y^{3}$ out of $y^{3} \cos^{2} (x)$ .

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{4}{y^{3}} (y^{3} (\cos^{2} (x)))$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y^{3}} \frac{d y}{d x} = \frac{4}{y^{3}} (y^{3} \cos^{2} (x))$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y^{3}} \frac{d y}{d x} = 4 \cos^{2} (x)$

Step 1.3

Rewrite the equation.

$\frac{1}{y^{3}} d y = 4 \cos^{2} (x) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{3}} d y = \int 4 \cos^{2} (x) d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1.1

Move $y^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{3})}^{- 1} d y = \int 4 \cos^{2} (x) d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{3})}^{- 1}$ .

Tap for more steps...

Step 2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int y^{3 \cdot - 1} d y = \int 4 \cos^{2} (x) d x$

Step 2.2.1.2.2

Multiply $3$ by $- 1$ .

$\int y^{- 3} d y = \int 4 \cos^{2} (x) d x$

Step 2.2.2

By the Power Rule, the integral of $y^{- 3}$ with respect to $y$ is $- \frac{1}{2} y^{- 2}$ .

$- \frac{1}{2} y^{- 2} + C_{1} = \int 4 \cos^{2} (x) d x$

Step 2.2.3

Simplify the answer.

Tap for more steps...

Step 2.2.3.1

Rewrite $- \frac{1}{2} y^{- 2} + C_{1}$ as $- \frac{1}{2} \cdot \frac{1}{y^{2}} + C_{1}$ .

$- \frac{1}{2} \cdot \frac{1}{y^{2}} + C_{1} = \int 4 \cos^{2} (x) d x$

Step 2.2.3.2

Simplify.

Tap for more steps...

Step 2.2.3.2.1

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

$- \frac{1}{y^{2} \cdot 2} + C_{1} = \int 4 \cos^{2} (x) d x$

Step 2.2.3.2.2

Move $2$ to the left of $y^{2}$ .

$- \frac{1}{2 y^{2}} + C_{1} = \int 4 \cos^{2} (x) d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$- \frac{1}{2 y^{2}} + C_{1} = 4 \int \cos^{2} (x) d x$

Step 2.3.2

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

$- \frac{1}{2 y^{2}} + C_{1} = 4 \int \frac{1 + \cos (2 x)}{2} d x$

Step 2.3.3

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

$- \frac{1}{2 y^{2}} + C_{1} = 4 (\frac{1}{2} \int 1 + \cos (2 x) d x)$

Step 2.3.4

Simplify.

Tap for more steps...

Step 2.3.4.1

Combine $\frac{1}{2}$ and $4$ .

$- \frac{1}{2 y^{2}} + C_{1} = \frac{4}{2} \int 1 + \cos (2 x) d x$

Step 2.3.4.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.3.4.2.1

Factor $2$ out of $4$ .

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

Step 2.3.4.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.4.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.4.2.2.2

Cancel the common factor.

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

Step 2.3.4.2.2.3

Rewrite the expression.

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

Step 2.3.4.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

Apply the constant rule.

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

Step 2.3.7

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

Tap for more steps...

Step 2.3.7.1

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

Tap for more steps...

Step 2.3.7.1.1

Differentiate $2 x$ .

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

Step 2.3.7.1.2

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 2.3.7.1.3

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 2.3.7.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.7.2

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

$- \frac{1}{2 y^{2}} + C_{1} = 2 (x + C_{2} + \frac{1}{2} (\sin (u) + C_{3}))$

Step 2.3.11

Simplify.

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

Step 2.3.12

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

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

Step 2.3.13

Simplify.

Tap for more steps...

Step 2.3.13.1

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

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

Step 2.3.13.2

Apply the distributive property.

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

Step 2.3.13.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.13.3.1

Cancel the common factor.

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

Step 2.3.13.3.2

Rewrite the expression.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 y^{2}, 1, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$2 y^{2}$

Step 3.2

Multiply each term in $- \frac{1}{2 y^{2}} = 2 x + \sin (2 x) + K$ by $2 y^{2}$ to eliminate the fractions.

Tap for more steps...

Step 3.2.1

Multiply each term in $- \frac{1}{2 y^{2}} = 2 x + \sin (2 x) + K$ by $2 y^{2}$ .

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

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

Tap for more steps...

Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{2 y^{2}}$ into the numerator.

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Simplify each term.

Tap for more steps...

Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.3.1.2

Multiply $2$ by $2$ .

$- 1 = 4 x y^{2} + \sin (2 x) (2 y^{2}) + K (2 y^{2})$

Step 3.2.3.1.3

Rewrite using the commutative property of multiplication.

$- 1 = 4 x y^{2} + 2 \sin (2 x) y^{2} + K (2 y^{2})$

Step 3.2.3.1.4

Rewrite using the commutative property of multiplication.

$- 1 = 4 x y^{2} + 2 \sin (2 x) y^{2} + 2 K y^{2}$

Step 3.2.3.2

Reorder factors in $4 x y^{2} + 2 \sin (2 x) y^{2} + 2 K y^{2}$ .

$- 1 = 4 x y^{2} + 2 y^{2} \sin (2 x) + 2 K y^{2}$

Step 3.3

Solve the equation.

Tap for more steps...

Step 3.3.1

Rewrite the equation as $4 x y^{2} + 2 y^{2} \sin (2 x) + 2 K y^{2} = - 1$ .

$4 x y^{2} + 2 y^{2} \sin (2 x) + 2 K y^{2} = - 1$

Step 3.3.2

Factor $2 y^{2}$ out of $4 x y^{2} + 2 y^{2} \sin (2 x) + 2 K y^{2}$ .

Tap for more steps...

Step 3.3.2.1

Factor $2 y^{2}$ out of $4 x y^{2}$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Factor $2 y^{2}$ out of $2 K y^{2}$ .

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

Step 3.3.2.4

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

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

Step 3.3.2.5

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

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

Step 3.3.3

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

Tap for more steps...

Step 3.3.3.1

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

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

Step 3.3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.3.2.1.1

Cancel the common factor.

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

Step 3.3.3.2.1.2

Rewrite the expression.

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

Step 3.3.3.2.2

Cancel the common factor of $2 x + \sin (2 x) + K$ .

Tap for more steps...

Step 3.3.3.2.2.1

Cancel the common factor.

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

Step 3.3.3.2.2.2

Divide $y^{2}$ by $1$ .

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

Step 3.3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.3.1

Move the negative in front of the fraction.

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

Step 3.3.4

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

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

Step 3.3.5

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

Tap for more steps...

Step 3.3.5.1

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

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

Step 3.3.5.2

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