Calculus Examples

$y \frac{d y}{d x} = x^{2} - 7 x$

Step 1

Rewrite the equation.

$y d y = (x^{2} - 7 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int x^{2} - 7 x d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int x^{2} - 7 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\frac{1}{2} y^{2} + C_{1} = \int x^{2} d x + \int - 7 x d x$

Step 2.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} + C_{2} + \int - 7 x d x$

Step 2.3.3

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} + C_{2} - 7 \int x d x$

Step 2.3.4

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} + C_{2} - 7 (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} - 7 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $- 7$ and $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} + \frac{- 7}{2} x^{2} + C_{4}$

Step 2.3.5.2.2

Move the negative in front of the fraction.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3} x^{3} - \frac{7}{2} x^{2} + C_{4}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (\frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{3} x^{3} - \frac{7}{2} x^{2} + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

$y^{2} = 2 (\frac{x^{3}}{3} - \frac{7}{2} x^{2} + K)$

Step 3.2.2.1.1.2

Combine $x^{2}$ and $\frac{7}{2}$ .

$y^{2} = 2 (\frac{x^{3}}{3} - \frac{x^{2} \cdot 7}{2} + K)$

Step 3.2.2.1.1.3

Move $7$ to the left of $x^{2}$ .

$y^{2} = 2 (\frac{x^{3}}{3} - \frac{7 x^{2}}{2} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{3}}{3} + 2 (- \frac{7 x^{2}}{2}) + 2 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Combine $2$ and $\frac{x^{3}}{3}$ .

$y^{2} = \frac{2 x^{3}}{3} + 2 (- \frac{7 x^{2}}{2}) + 2 K$

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Move the leading negative in $- \frac{7 x^{2}}{2}$ into the numerator.

$y^{2} = \frac{2 x^{3}}{3} + 2 \frac{- 7 x^{2}}{2} + 2 K$

Step 3.2.2.1.3.2.2

Cancel the common factor.

$y^{2} = \frac{2 x^{3}}{3} + 2 \frac{- 7 x^{2}}{2} + 2 K$

Step 3.2.2.1.3.2.3

Rewrite the expression.

$y^{2} = \frac{2 x^{3}}{3} - 7 x^{2} + 2 K$

Step 3.3

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

$y = \pm \sqrt{\frac{2 x^{3}}{3} - 7 x^{2} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{\frac{2 x^{3}}{3} - 7 x^{2} + 2 K}$ .

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Step 3.4.1

To write $- 7 x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{2 x^{3}}{3} - 7 x^{2} \cdot \frac{3}{3} + 2 K}$

Step 3.4.2

Simplify terms.

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Step 3.4.2.1

Combine $- 7 x^{2}$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{2 x^{3}}{3} + \frac{- 7 x^{2} \cdot 3}{3} + 2 K}$

Step 3.4.2.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{2 x^{3} - 7 x^{2} \cdot 3}{3} + 2 K}$

Step 3.4.3

Simplify the numerator.

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Step 3.4.3.1

Factor $x^{2}$ out of $2 x^{3} - 7 x^{2} \cdot 3$ .

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Step 3.4.3.1.1

Factor $x^{2}$ out of $2 x^{3}$ .

$y = \pm \sqrt{\frac{x^{2} (2 x) - 7 x^{2} \cdot 3}{3} + 2 K}$

Step 3.4.3.1.2

Factor $x^{2}$ out of $- 7 x^{2} \cdot 3$ .

$y = \pm \sqrt{\frac{x^{2} (2 x) + x^{2} (- 7 \cdot 3)}{3} + 2 K}$

Step 3.4.3.1.3

Factor $x^{2}$ out of $x^{2} (2 x) + x^{2} (- 7 \cdot 3)$ .

$y = \pm \sqrt{\frac{x^{2} (2 x - 7 \cdot 3)}{3} + 2 K}$

Step 3.4.3.2

Multiply $- 7$ by $3$ .

$y = \pm \sqrt{\frac{x^{2} (2 x - 21)}{3} + 2 K}$

Step 3.4.4

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x^{2} (2 x - 21)}{3} + 2 K \cdot \frac{3}{3}}$

Step 3.4.5

Simplify terms.

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Step 3.4.5.1

Combine $2 K$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{x^{2} (2 x - 21)}{3} + \frac{2 K \cdot 3}{3}}$

Step 3.4.5.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} (2 x - 21) + 2 K \cdot 3}{3}}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Apply the distributive property.

$y = \pm \sqrt{\frac{x^{2} (2 x) + x^{2} \cdot - 21 + 2 K \cdot 3}{3}}$

Step 3.4.6.2

Rewrite using the commutative property of multiplication.

$y = \pm \sqrt{\frac{2 x^{2} x + x^{2} \cdot - 21 + 2 K \cdot 3}{3}}$

Step 3.4.6.3

Move $- 21$ to the left of $x^{2}$ .

$y = \pm \sqrt{\frac{2 x^{2} x - 21 \cdot x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.6.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.4.6.4.1

Move $x$ .

$y = \pm \sqrt{\frac{2 (x \cdot x^{2}) - 21 \cdot x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.6.4.2

Multiply $x$ by $x^{2}$ .

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Step 3.4.6.4.2.1

Raise $x$ to the power of $1$ .

$y = \pm \sqrt{\frac{2 (x^{1} x^{2}) - 21 \cdot x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.6.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \sqrt{\frac{2 x^{1 + 2} - 21 \cdot x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.6.4.3

Add $1$ and $2$ .

$y = \pm \sqrt{\frac{2 x^{3} - 21 \cdot x^{2} + 2 K \cdot 3}{3}}$

$y = \pm \sqrt{\frac{2 x^{3} - 21 x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.6.5

Multiply $3$ by $2$ .

$y = \pm \sqrt{\frac{2 x^{3} - 21 x^{2} + 6 K}{3}}$

Step 3.4.7

Rewrite $\sqrt{\frac{2 x^{3} - 21 x^{2} + 6 K}{3}}$ as $\frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K}}{\sqrt{3}}$

Step 3.4.8

Multiply $\frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4.9

Combine and simplify the denominator.

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Step 3.4.9.1

Multiply $\frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.9.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.9.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.9.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.9.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.9.6.2

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

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.9.6.3

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

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.9.6.4

Cancel the common factor of $2$ .

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Step 3.4.9.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.9.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{3^{1}}$

Step 3.4.9.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2 x^{3} - 21 x^{2} + 6 K} \sqrt{3}}{3}$

Step 3.4.10

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(2 x^{3} - 21 x^{2} + 6 K) \cdot 3}}{3}$

Step 3.4.11

Reorder factors in $\pm \frac{\sqrt{(2 x^{3} - 21 x^{2} + 6 K) \cdot 3}}{3}$ .

$y = \pm \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

Step 3.5

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

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Step 3.5.1

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

$y = \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

Step 3.5.2

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

$y = - \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

Step 3.5.3

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

$y = \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

$y = - \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

$y = \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

$y = - \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

$y = \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

$y = - \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + 6 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + K)}}{3}$

$y = - \frac{\sqrt{3 (2 x^{3} - 21 x^{2} + K)}}{3}$