Calculus Examples

$(15 x^{2} - 14 x) d x - d y = 0$

Step 1

Subtract $(15 x^{2} - 14 x) d x$ from both sides of the equation.

$- d y = - (15 x^{2} - 14 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int - 1 d y = \int - (15 x^{2} - 14 x) d x$

Step 2.2

Apply the constant rule.

$- y + C_{1} = \int - (15 x^{2} - 14 x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$- y + C_{1} = - \int 15 x^{2} - 14 x d x$

Step 2.3.2

Split the single integral into multiple integrals.

$- y + C_{1} = - (\int 15 x^{2} d x + \int - 14 x d x)$

Step 2.3.3

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$- y + C_{1} = - (15 \int x^{2} d x + \int - 14 x d x)$

Step 2.3.4

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

$- y + C_{1} = - (15 (\frac{1}{3} x^{3} + C_{2}) + \int - 14 x d x)$

Step 2.3.5

Since $- 14$ is constant with respect to $x$ , move $- 14$ out of the integral.

$- y + C_{1} = - (15 (\frac{1}{3} x^{3} + C_{2}) - 14 \int x d x)$

Step 2.3.6

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

$- y + C_{1} = - (15 (\frac{1}{3} x^{3} + C_{2}) - 14 (\frac{1}{2} x^{2} + C_{3}))$

Step 2.3.7

Simplify.

$- y + C_{1} = - (5 x^{3} - 7 x^{2}) + C_{4}$

Step 2.4

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

$- y = - (5 x^{3} - 7 x^{2}) + K$

Step 3

Divide each term in $- y = - (5 x^{3} - 7 x^{2}) + K$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- y = - (5 x^{3} - 7 x^{2}) + K$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- (5 x^{3} - 7 x^{2})}{- 1} + \frac{K}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- (5 x^{3} - 7 x^{2})}{- 1} + \frac{K}{- 1}$

Step 3.2.2

Divide $y$ by $1$ .

$y = \frac{- (5 x^{3} - 7 x^{2})}{- 1} + \frac{K}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{5 x^{3} - 7 x^{2}}{1} + \frac{K}{- 1}$

Step 3.3.1.2

Divide $5 x^{3} - 7 x^{2}$ by $1$ .

$y = 5 x^{3} - 7 x^{2} + \frac{K}{- 1}$

Step 3.3.1.3

Move the negative one from the denominator of $\frac{K}{- 1}$ .

$y = 5 x^{3} - 7 x^{2} - 1 \cdot K$

Step 3.3.1.4

Rewrite $- 1 \cdot K$ as $- K$ .

$y = 5 x^{3} - 7 x^{2} - K$

Step 4

Simplify the constant of integration.

$y = 5 x^{3} - 7 x^{2} + K$