Calculus Examples

$\frac{d y}{d x} = 9 x^{2} - 4 x + 5$ , $y (- 1) = 0$

Step 1

Rewrite the equation.

$d y = (9 x^{2} - 4 x + 5) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 9 x^{2} - 4 x + 5 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 9 x^{2} - 4 x + 5 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 9 x^{2} d x + \int - 4 x d x + \int 5 d x$

Step 2.3.2

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$y + C_{1} = 9 \int x^{2} d x + \int - 4 x d x + \int 5 d x$

Step 2.3.3

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

$y + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2}) + \int - 4 x d x + \int 5 d x$

Step 2.3.4

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

$y + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2}) - 4 \int x d x + \int 5 d x$

Step 2.3.5

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

$y + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + \int 5 d x$

Step 2.3.6

Apply the constant rule.

$y + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + 5 x + C_{4}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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Step 2.3.7.1.1

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

$y + C_{1} = 9 (\frac{x^{3}}{3} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + 5 x + C_{4}$

Step 2.3.7.1.2

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

$y + C_{1} = 9 (\frac{x^{3}}{3} + C_{2}) - 4 (\frac{x^{2}}{2} + C_{3}) + 5 x + C_{4}$

Step 2.3.7.2

Simplify.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $- 1$ for $x$ and $0$ for $y$ in $y = 3 x^{3} - 2 x^{2} + 5 x + K$ .

$0 = 3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 \cdot - 1 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 \cdot - 1 + K = 0$ .

$3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 \cdot - 1 + K = 0$

Step 4.2

Simplify $3 {(- 1)}^{3} - 2 {(- 1)}^{2} + 5 \cdot - 1 + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 1$ to the power of $3$ .

$3 \cdot - 1 - 2 {(- 1)}^{2} + 5 \cdot - 1 + K = 0$

Step 4.2.1.2

Multiply $3$ by $- 1$ .

$- 3 - 2 {(- 1)}^{2} + 5 \cdot - 1 + K = 0$

Step 4.2.1.3

Raise $- 1$ to the power of $2$ .

$- 3 - 2 \cdot 1 + 5 \cdot - 1 + K = 0$

Step 4.2.1.4

Multiply $- 2$ by $1$ .

$- 3 - 2 + 5 \cdot - 1 + K = 0$

Step 4.2.1.5

Multiply $5$ by $- 1$ .

$- 3 - 2 - 5 + K = 0$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $2$ from $- 3$ .

$- 5 - 5 + K = 0$

Step 4.2.2.2

Subtract $5$ from $- 5$ .

$- 10 + K = 0$

Step 4.3

Add $10$ to both sides of the equation.

$K = 10$

Step 5

Substitute $10$ for $K$ in $y = 3 x^{3} - 2 x^{2} + 5 x + K$ and simplify.

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Step 5.1

Substitute $10$ for $K$ .

$y = 3 x^{3} - 2 x^{2} + 5 x + 10$