Calculus Examples

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

Step 1

Rewrite the equation.

$d y = (x^{2} - 2 x + 3) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int x^{2} - 2 x + 3 d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

Tap for more steps...

Step 2.3.6.1

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

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

Step 2.3.6.2

Simplify.

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

Step 2.3.6.3

Reorder terms.

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $4$ for $y$ in $y = - x^{2} + \frac{1}{3} x^{3} + 3 x + K$ .

$4 = - 1^{2} + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $- 1^{2} + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K = 4$ .

$- 1^{2} + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K = 4$

Step 4.2

Simplify $- 1^{2} + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

One to any power is one.

$- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K = 4$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$- 1 + \frac{1}{3} \cdot 1^{3} + 3 \cdot 1 + K = 4$

Step 4.2.1.3

One to any power is one.

$- 1 + \frac{1}{3} \cdot 1 + 3 \cdot 1 + K = 4$

Step 4.2.1.4

Multiply $\frac{1}{3}$ by $1$ .

$- 1 + \frac{1}{3} + 3 \cdot 1 + K = 4$

Step 4.2.1.5

Multiply $3$ by $1$ .

$- 1 + \frac{1}{3} + 3 + K = 4$

Step 4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 1 \cdot \frac{3}{3} + \frac{1}{3} + 3 + K = 4$

Step 4.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{- 1 \cdot 3}{3} + \frac{1}{3} + 3 + K = 4$

Step 4.2.4

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 3 + 1}{3} + 3 + K = 4$

Step 4.2.5

Simplify the numerator.

Tap for more steps...

Step 4.2.5.1

Multiply $- 1$ by $3$ .

$\frac{- 3 + 1}{3} + 3 + K = 4$

Step 4.2.5.2

Add $- 3$ and $1$ .

$\frac{- 2}{3} + 3 + K = 4$

Step 4.2.6

Move the negative in front of the fraction.

$- \frac{2}{3} + 3 + K = 4$

Step 4.2.7

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

$- \frac{2}{3} + 3 \cdot \frac{3}{3} + K = 4$

Step 4.2.8

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

$- \frac{2}{3} + \frac{3 \cdot 3}{3} + K = 4$

Step 4.2.9

Combine the numerators over the common denominator.

$\frac{- 2 + 3 \cdot 3}{3} + K = 4$

Step 4.2.10

Simplify the numerator.

Tap for more steps...

Step 4.2.10.1

Multiply $3$ by $3$ .

$\frac{- 2 + 9}{3} + K = 4$

Step 4.2.10.2

Add $- 2$ and $9$ .

$\frac{7}{3} + K = 4$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

Tap for more steps...

Step 4.3.1

Subtract $\frac{7}{3}$ from both sides of the equation.

$K = 4 - \frac{7}{3}$

Step 4.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$K = 4 \cdot \frac{3}{3} - \frac{7}{3}$

Step 4.3.3

Combine $4$ and $\frac{3}{3}$ .

$K = \frac{4 \cdot 3}{3} - \frac{7}{3}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{4 \cdot 3 - 7}{3}$

Step 4.3.5

Simplify the numerator.

Tap for more steps...

Step 4.3.5.1

Multiply $4$ by $3$ .

$K = \frac{12 - 7}{3}$

Step 4.3.5.2

Subtract $7$ from $12$ .

$K = \frac{5}{3}$

Step 5

Substitute $\frac{5}{3}$ for $K$ in $y = - x^{2} + \frac{1}{3} x^{3} + 3 x + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $\frac{5}{3}$ for $K$ .

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

Step 5.2

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

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