Calculus Examples

$\frac{d y}{d x} = 10 x - 9$ and $y (5) = 0$

Step 1

Rewrite the equation.

$d y = (10 x - 9) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 10 x - 9 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 10 x - 9 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 10 x d x + \int - 9 d x$

Step 2.3.2

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$y + C_{1} = 10 \int x d x + \int - 9 d x$

Step 2.3.3

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

$y + C_{1} = 10 (\frac{1}{2} x^{2} + C_{2}) + \int - 9 d x$

Step 2.3.4

Apply the constant rule.

$y + C_{1} = 10 (\frac{1}{2} x^{2} + C_{2}) - 9 x + C_{3}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$y + C_{1} = 10 (\frac{x^{2}}{2} + C_{2}) - 9 x + C_{3}$

Step 2.3.5.2

Simplify.

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

Step 2.4

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

$y = 5 x^{2} - 9 x + K$

Step 3

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

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

Step 4

Solve for $K$ .

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $5$ by $5^{2}$ by adding the exponents.

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Step 4.2.1.1.1

Multiply $5$ by $5^{2}$ .

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Step 4.2.1.1.1.1

Raise $5$ to the power of $1$ .

$5^{1} \cdot 5^{2} - 9 \cdot 5 + K = 0$

Step 4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{1 + 2} - 9 \cdot 5 + K = 0$

Step 4.2.1.1.2

Add $1$ and $2$ .

$5^{3} - 9 \cdot 5 + K = 0$

Step 4.2.1.2

Raise $5$ to the power of $3$ .

$125 - 9 \cdot 5 + K = 0$

Step 4.2.1.3

Multiply $- 9$ by $5$ .

$125 - 45 + K = 0$

Step 4.2.2

Subtract $45$ from $125$ .

$80 + K = 0$

Step 4.3

Subtract $80$ from both sides of the equation.

$K = - 80$

Step 5

Substitute $- 80$ for $K$ in $y = 5 x^{2} - 9 x + K$ and simplify.

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Step 5.1

Substitute $- 80$ for $K$ .

$y = 5 x^{2} - 9 x - 80$