Calculus Examples

$x^{2} \frac{d y}{d x} = 6 x^{4} + 5 x + 5$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{2} \frac{d y}{d x} = 6 x^{4} + 5 x + 5$ by $x^{2}$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} \frac{d y}{d x} = 6 x^{4} + 5 x + 5$ by $x^{2}$ .

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{6 x^{4}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d y}{d x}}{x^{2}} = \frac{6 x^{4}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{6 x^{4}}{x^{2}} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 1.1.3.1.1.1

Factor $x^{2}$ out of $6 x^{4}$ .

$\frac{d y}{d x} = \frac{x^{2} (6 x^{2})}{x^{2}} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Multiply by $1$ .

$\frac{d y}{d x} = \frac{x^{2} (6 x^{2})}{x^{2} \cdot 1} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x^{2} (6 x^{2})}{x^{2} \cdot 1} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{6 x^{2}}{1} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.1.2.4

Divide $6 x^{2}$ by $1$ .

$\frac{d y}{d x} = 6 x^{2} + \frac{5 x}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Factor $x$ out of $5 x$ .

$\frac{d y}{d x} = 6 x^{2} + \frac{x \cdot 5}{x^{2}} + \frac{5}{x^{2}}$

Step 1.1.3.1.2.2

Cancel the common factors.

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Step 1.1.3.1.2.2.1

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

$\frac{d y}{d x} = 6 x^{2} + \frac{x \cdot 5}{x \cdot x} + \frac{5}{x^{2}}$

Step 1.1.3.1.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = 6 x^{2} + \frac{x \cdot 5}{x \cdot x} + \frac{5}{x^{2}}$

Step 1.1.3.1.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = 6 x^{2} + \frac{5}{x} + \frac{5}{x^{2}}$

Step 1.2

Rewrite the equation.

$d y = (6 x^{2} + \frac{5}{x} + \frac{5}{x^{2}}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 6 x^{2} + \frac{5}{x} + \frac{5}{x^{2}} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 6 x^{2} + \frac{5}{x} + \frac{5}{x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 6 x^{2} d x + \int \frac{5}{x} d x + \int \frac{5}{x^{2}} d x$

Step 2.3.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$y + C_{1} = 6 \int x^{2} d x + \int \frac{5}{x} d x + \int \frac{5}{x^{2}} d x$

Step 2.3.3

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

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + \int \frac{5}{x} d x + \int \frac{5}{x^{2}} d x$

Step 2.3.4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 5 \int \frac{1}{x} d x + \int \frac{5}{x^{2}} d x$

Step 2.3.5

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

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + \int \frac{5}{x^{2}} d x$

Step 2.3.6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 \int \frac{1}{x^{2}} d x$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 \int {(x^{2})}^{- 1} d x$

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$y + C_{1} = 6 (\frac{x^{3}}{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 \int {(x^{2})}^{- 1} d x$

Step 2.3.7.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.7.2.2.1

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

$y + C_{1} = 6 (\frac{x^{3}}{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 \int x^{2 \cdot - 1} d x$

Step 2.3.7.2.2.2

Multiply $2$ by $- 1$ .

$y + C_{1} = 6 (\frac{x^{3}}{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 \int x^{- 2} d x$

Step 2.3.8

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

$y + C_{1} = 6 (\frac{x^{3}}{3} + C_{2}) + 5 (\ln (| x |) + C_{3}) + 5 (- x^{- 1} + C_{4})$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Simplify.

$y + C_{1} = 2 x^{3} + 5 \ln (| x |) + 5 (- x^{- 1}) + C_{5}$

Step 2.3.9.2

Multiply $- 1$ by $5$ .

$y + C_{1} = 2 x^{3} + 5 \ln (| x |) - 5 x^{- 1} + C_{5}$

Step 2.3.10

Reorder terms.

$y + C_{1} = 2 x^{3} - 5 x^{- 1} + 5 \ln (| x |) + C_{5}$

Step 2.4

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

$y = 2 x^{3} - 5 x^{- 1} + 5 \ln (| x |) + C$