Calculus Examples

$\frac{d y}{d x} = 8 x - 5$ and $y (9) = 0$

Step 1

Rewrite the equation.

$d y = (8 x - 5) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 8 x - 5 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 8 x - 5 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 8 x d x + \int - 5 d x$

Step 2.3.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$y + C_{1} = 8 \int x d x + \int - 5 d x$

Step 2.3.3

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

$y + C_{1} = 8 (\frac{1}{2} x^{2} + C_{2}) + \int - 5 d x$

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

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

Step 2.4

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

$y = 4 x^{2} - 5 x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $9$ for $x$ and $0$ for $y$ in $y = 4 x^{2} - 5 x + K$ .

$0 = 4 \cdot 9^{2} - 5 \cdot 9 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $4 \cdot 9^{2} - 5 \cdot 9 + K = 0$ .

$4 \cdot 9^{2} - 5 \cdot 9 + K = 0$

Step 4.2

Simplify $4 \cdot 9^{2} - 5 \cdot 9 + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $9$ to the power of $2$ .

$4 \cdot 81 - 5 \cdot 9 + K = 0$

Step 4.2.1.2

Multiply $4$ by $81$ .

$324 - 5 \cdot 9 + K = 0$

Step 4.2.1.3

Multiply $- 5$ by $9$ .

$324 - 45 + K = 0$

Step 4.2.2

Subtract $45$ from $324$ .

$279 + K = 0$

Step 4.3

Subtract $279$ from both sides of the equation.

$K = - 279$

Step 5

Substitute $- 279$ for $K$ in $y = 4 x^{2} - 5 x + K$ and simplify.

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Step 5.1

Substitute $- 279$ for $K$ .

$y = 4 x^{2} - 5 x - 279$