Calculus Examples

$\frac{d y}{d x} = 4 x^{3} - 3 x^{2} - 24 x + 8$ ; cuando $f (3) = - 6$

Step 1

Rewrite the equation.

$d y = (4 x^{3} - 3 x^{2} - 24 x + 8) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 4 x^{3} - 3 x^{2} - 24 x + 8 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 x^{3} - 3 x^{2} - 24 x + 8 d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 4 x^{3} d x + \int - 3 x^{2} d x + \int - 24 x d x + \int 8 d x$

Step 2.3.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int x^{3} d x + \int - 3 x^{2} d x + \int - 24 x d x + \int 8 d x$

Step 2.3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) + \int - 3 x^{2} d x + \int - 24 x d x + \int 8 d x$

Step 2.3.4

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

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 \int x^{2} d x + \int - 24 x d x + \int 8 d x$

Step 2.3.5

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

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) + \int - 24 x d x + \int 8 d x$

Step 2.3.6

Since $- 24$ is constant with respect to $x$ , move $- 24$ out of the integral.

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) - 24 \int x d x + \int 8 d x$

Step 2.3.7

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

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) - 24 (\frac{1}{2} x^{2} + C_{4}) + \int 8 d x$

Step 2.3.8

Apply the constant rule.

$y + C_{1} = 4 (\frac{1}{4} x^{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) - 24 (\frac{1}{2} x^{2} + C_{4}) + 8 x + C_{5}$

Step 2.3.9

Simplify.

Tap for more steps...

Step 2.3.9.1

Simplify.

Tap for more steps...

Step 2.3.9.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y + C_{1} = 4 (\frac{x^{4}}{4} + C_{2}) - 3 (\frac{1}{3} x^{3} + C_{3}) - 24 (\frac{1}{2} x^{2} + C_{4}) + 8 x + C_{5}$

Step 2.3.9.1.2

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

$y + C_{1} = 4 (\frac{x^{4}}{4} + C_{2}) - 3 (\frac{x^{3}}{3} + C_{3}) - 24 (\frac{1}{2} x^{2} + C_{4}) + 8 x + C_{5}$

Step 2.3.9.1.3

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

$y + C_{1} = 4 (\frac{x^{4}}{4} + C_{2}) - 3 (\frac{x^{3}}{3} + C_{3}) - 24 (\frac{x^{2}}{2} + C_{4}) + 8 x + C_{5}$

Step 2.3.9.2

Simplify.

$y + C_{1} = x^{4} - x^{3} - 12 x^{2} + 8 x + C_{6}$

Step 2.4

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

$y = x^{4} - x^{3} - 12 x^{2} + 8 x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $3$ for $x$ and $- 6$ for $y$ in $y = x^{4} - x^{3} - 12 x^{2} + 8 x + K$ .

$- 6 = 3^{4} - 3^{3} - 12 \cdot 3^{2} + 8 \cdot 3 + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $3^{4} - 3^{3} - 12 \cdot 3^{2} + 8 \cdot 3 + K = - 6$ .

$3^{4} - 3^{3} - 12 \cdot 3^{2} + 8 \cdot 3 + K = - 6$

Step 4.2

Simplify $3^{4} - 3^{3} - 12 \cdot 3^{2} + 8 \cdot 3 + K$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $3$ to the power of $4$ .

$81 - 3^{3} - 12 \cdot 3^{2} + 8 \cdot 3 + K = - 6$

Step 4.2.1.2

Raise $3$ to the power of $3$ .

$81 - 1 \cdot 27 - 12 \cdot 3^{2} + 8 \cdot 3 + K = - 6$

Step 4.2.1.3

Multiply $- 1$ by $27$ .

$81 - 27 - 12 \cdot 3^{2} + 8 \cdot 3 + K = - 6$

Step 4.2.1.4

Raise $3$ to the power of $2$ .

$81 - 27 - 12 \cdot 9 + 8 \cdot 3 + K = - 6$

Step 4.2.1.5

Multiply $- 12$ by $9$ .

$81 - 27 - 108 + 8 \cdot 3 + K = - 6$

Step 4.2.1.6

Multiply $8$ by $3$ .

$81 - 27 - 108 + 24 + K = - 6$

Step 4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.2.1

Subtract $27$ from $81$ .

$54 - 108 + 24 + K = - 6$

Step 4.2.2.2

Subtract $108$ from $54$ .

$- 54 + 24 + K = - 6$

Step 4.2.2.3

Add $- 54$ and $24$ .

$- 30 + K = - 6$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Add $30$ to both sides of the equation.

$K = - 6 + 30$

Step 4.3.2

Add $- 6$ and $30$ .

$K = 24$

Step 5

Substitute $24$ for $K$ in $y = x^{4} - x^{3} - 12 x^{2} + 8 x + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $24$ for $K$ .

$y = x^{4} - x^{3} - 12 x^{2} + 8 x + 24$