Calculus Examples

$(2 x - 1) d x = - (3 y + 7) d y$

Step 1

Rewrite the equation.

$- (3 y + 7) d y = (2 x - 1) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int - (3 y + 7) d y = \int 2 x - 1 d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Multiply $- (3 y + 7)$ .

$\int - (3 y) - 1 \cdot 7 d y = \int 2 x - 1 d x$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Multiply $3$ by $- 1$ .

$\int - 3 y - 1 \cdot 7 d y = \int 2 x - 1 d x$

Step 2.2.2.2

Multiply $- 1$ by $7$ .

$\int - 3 y - 7 d y = \int 2 x - 1 d x$

Step 2.2.3

Split the single integral into multiple integrals.

$\int - 3 y d y + \int - 7 d y = \int 2 x - 1 d x$

Step 2.2.4

Since $- 3$ is constant with respect to $y$ , move $- 3$ out of the integral.

$- 3 \int y d y + \int - 7 d y = \int 2 x - 1 d x$

Step 2.2.5

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

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

Step 2.2.6

Apply the constant rule.

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

Step 2.2.7

Simplify.

Tap for more steps...

Step 2.2.7.1

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

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

Step 2.2.7.2

Simplify.

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

Step 2.2.7.3

Move the negative in front of the fraction.

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

Step 2.2.8

Reorder terms.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

$- \frac{3}{2} y^{2} - 7 y + C_{3} = x^{2} - x + C_{6}$

Step 2.4

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

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Move $3$ to the left of $y^{2}$ .

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

Step 3.2

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 3.2.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 3.2.2

Add $x$ to both sides of the equation.

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

Step 3.2.3

Subtract $K$ from both sides of the equation.

$- \frac{3 y^{2}}{2} - 7 y - x^{2} + x - K = 0$

Step 3.3

Multiply through by the least common denominator $2$ , then simplify.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

$2 (- \frac{3 y^{2}}{2}) + 2 (- 7 y) + 2 (- x^{2}) + 2 x + 2 (- K) = 0$

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Move the leading negative in $- \frac{3 y^{2}}{2}$ into the numerator.

$2 (\frac{- 3 y^{2}}{2}) + 2 (- 7 y) + 2 (- x^{2}) + 2 x + 2 (- K) = 0$

Step 3.3.2.1.2

Cancel the common factor.

$2 (\frac{- 3 y^{2}}{2}) + 2 (- 7 y) + 2 (- x^{2}) + 2 x + 2 (- K) = 0$

Step 3.3.2.1.3

Rewrite the expression.

$- 3 y^{2} + 2 (- 7 y) + 2 (- x^{2}) + 2 x + 2 (- K) = 0$

Step 3.3.2.2

Multiply $- 7$ by $2$ .

$- 3 y^{2} - 14 y + 2 (- x^{2}) + 2 x + 2 (- K) = 0$

Step 3.3.2.3

Multiply $- 1$ by $2$ .

$- 3 y^{2} - 14 y - 2 x^{2} + 2 x + 2 (- K) = 0$

Step 3.3.2.4

Multiply $- 1$ by $2$ .

$- 3 y^{2} - 14 y - 2 x^{2} + 2 x - 2 K = 0$

Step 3.3.3

Move $- 14 y$ .

$- 3 y^{2} - 2 x^{2} + 2 x - 2 K - 14 y = 0$

Step 3.3.4

Move $2 x$ .

$- 3 y^{2} - 2 x^{2} - 2 K + 2 x - 14 y = 0$

Step 3.3.5

Reorder $- 3 y^{2}$ and $- 2 x^{2}$ .

$- 2 x^{2} - 3 y^{2} - 2 K + 2 x - 14 y = 0$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5

Substitute the values $a = - 3$ , $b = - 14$ , and $c = - 2 x^{2} - 2 K + 2 x$ into the quadratic formula and solve for $y$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (- 3 \cdot (- 2 x^{2} - 2 K + 2 x))}}{2 \cdot - 3}$

Step 3.6

Simplify.

Tap for more steps...

Step 3.6.1

Simplify the numerator.

Tap for more steps...

Step 3.6.1.1

Raise $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot - 3 \cdot (- 2 x^{2} - 2 K + 2 x)}}{2 \cdot - 3}$

Step 3.6.1.2

Multiply $- 4$ by $- 3$ .

$y = \frac{14 \pm \sqrt{196 + 12 \cdot (- 2 x^{2} - 2 K + 2 x)}}{2 \cdot - 3}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{14 \pm \sqrt{196 + 12 (- 2 x^{2}) + 12 (- 2 K) + 12 (2 x)}}{2 \cdot - 3}$

Step 3.6.1.4

Simplify.

Tap for more steps...

Step 3.6.1.4.1

Multiply $- 2$ by $12$ .

$y = \frac{14 \pm \sqrt{196 - 24 x^{2} + 12 (- 2 K) + 12 (2 x)}}{2 \cdot - 3}$

Step 3.6.1.4.2

Multiply $- 2$ by $12$ .

$y = \frac{14 \pm \sqrt{196 - 24 x^{2} - 24 K + 12 (2 x)}}{2 \cdot - 3}$

Step 3.6.1.4.3

Multiply $2$ by $12$ .

$y = \frac{14 \pm \sqrt{196 - 24 x^{2} - 24 K + 24 x}}{2 \cdot - 3}$

Step 3.6.1.5

Factor $4$ out of $196 - 24 x^{2} - 24 K + 24 x$ .

Tap for more steps...

Step 3.6.1.5.1

Factor $4$ out of $196$ .

$y = \frac{14 \pm \sqrt{4 (49) - 24 x^{2} - 24 K + 24 x}}{2 \cdot - 3}$

Step 3.6.1.5.2

Factor $4$ out of $- 24 x^{2}$ .

$y = \frac{14 \pm \sqrt{4 (49) + 4 (- 6 x^{2}) - 24 K + 24 x}}{2 \cdot - 3}$

Step 3.6.1.5.3

Factor $4$ out of $- 24 K$ .

$y = \frac{14 \pm \sqrt{4 (49) + 4 (- 6 x^{2}) + 4 (- 6 K) + 24 x}}{2 \cdot - 3}$

Step 3.6.1.5.4

Factor $4$ out of $24 x$ .

$y = \frac{14 \pm \sqrt{4 (49) + 4 (- 6 x^{2}) + 4 (- 6 K) + 4 (6 x)}}{2 \cdot - 3}$

Step 3.6.1.5.5

Factor $4$ out of $4 (49) + 4 (- 6 x^{2})$ .

$y = \frac{14 \pm \sqrt{4 (49 - 6 x^{2}) + 4 (- 6 K) + 4 (6 x)}}{2 \cdot - 3}$

Step 3.6.1.5.6

Factor $4$ out of $4 (49 - 6 x^{2}) + 4 (- 6 K)$ .

$y = \frac{14 \pm \sqrt{4 (49 - 6 x^{2} - 6 K) + 4 (6 x)}}{2 \cdot - 3}$

Step 3.6.1.5.7

Factor $4$ out of $4 (49 - 6 x^{2} - 6 K) + 4 (6 x)$ .

$y = \frac{14 \pm \sqrt{4 (49 - 6 x^{2} - 6 K + 6 x)}}{2 \cdot - 3}$

Step 3.6.1.6

Rewrite $4 (49 - 6 x^{2} - 6 K + 6 x)$ as $2^{2} (7^{2} - 6 x^{2} - 6 K + 6 x)$ .

Tap for more steps...

Step 3.6.1.6.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{14 \pm \sqrt{2^{2} (49 - 6 x^{2} - 6 K + 6 x)}}{2 \cdot - 3}$

Step 3.6.1.6.2

Rewrite $49$ as $7^{2}$ .

$y = \frac{14 \pm \sqrt{2^{2} (7^{2} - 6 x^{2} - 6 K + 6 x)}}{2 \cdot - 3}$

Step 3.6.1.7

Pull terms out from under the radical.

$y = \frac{14 \pm 2 \sqrt{7^{2} - 6 x^{2} - 6 K + 6 x}}{2 \cdot - 3}$

Step 3.6.1.8

Raise $7$ to the power of $2$ .

$y = \frac{14 \pm 2 \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{2 \cdot - 3}$

Step 3.6.2

Multiply $2$ by $- 3$ .

$y = \frac{14 \pm 2 \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{- 6}$

Step 3.6.3

Simplify $\frac{14 \pm 2 \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{- 6}$ .

$y = \frac{7 \pm \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{- 3}$

Step 3.6.4

Move the negative in front of the fraction.

$y = - \frac{7 \pm \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{3}$

Step 3.7

The final answer is the combination of both solutions.

$y = - \frac{7 + \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{3}$

$y = - \frac{7 - \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{3}$

$y = - \frac{7 + \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{3}$

$y = - \frac{7 - \sqrt{49 - 6 x^{2} - 6 K + 6 x}}{3}$

Step 4

Simplify the constant of integration.

$y = - \frac{7 + \sqrt{49 - 6 x^{2} + K + 6 x}}{3}$

$y = - \frac{7 - \sqrt{49 - 6 x^{2} + K + 6 x}}{3}$