Calculus Examples

$\frac{d y}{d x} = 6 x - 5$ and $y (8) = 0$

Step 1

Rewrite the equation.

$d y = (6 x - 5) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int 6 x - 5 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 6 x - 5 d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 6 x d x + \int - 5 d x$

Step 2.3.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$y + C_{1} = 6 \int x d x + \int - 5 d x$

Step 2.3.3

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

$y + C_{1} = 6 (\frac{1}{2} x^{2} + C_{2}) + \int - 5 d x$

Step 2.3.4

Apply the constant rule.

$y + C_{1} = 6 (\frac{1}{2} x^{2} + C_{2}) - 5 x + C_{3}$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

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

$y + C_{1} = 6 (\frac{x^{2}}{2} + C_{2}) - 5 x + C_{3}$

Step 2.3.5.2

Simplify.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $8$ for $x$ and $0$ for $y$ in $y = 3 x^{2} - 5 x + K$ .

$0 = 3 \cdot 8^{2} - 5 \cdot 8 + K$

Step 4

Solve for $K$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $3 \cdot 8^{2} - 5 \cdot 8 + K = 0$ .

$3 \cdot 8^{2} - 5 \cdot 8 + K = 0$

Step 4.2

Simplify $3 \cdot 8^{2} - 5 \cdot 8 + K$ .

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $8$ to the power of $2$ .

$3 \cdot 64 - 5 \cdot 8 + K = 0$

Step 4.2.1.2

Multiply $3$ by $64$ .

$192 - 5 \cdot 8 + K = 0$

Step 4.2.1.3

Multiply $- 5$ by $8$ .

$192 - 40 + K = 0$

Step 4.2.2

Subtract $40$ from $192$ .

$152 + K = 0$

Step 4.3

Subtract $152$ from both sides of the equation.

$K = - 152$

Step 5

Substitute $- 152$ for $K$ in $y = 3 x^{2} - 5 x + K$ and simplify.

Tap for more steps...

Step 5.1

Substitute $- 152$ for $K$ .

$y = 3 x^{2} - 5 x - 152$