Calculus Examples

$\frac{d y}{d x} = 6 x^{2} + 8 x$ ; , $y (3) = 11$

Step 1

Rewrite the equation.

$d y = (6 x^{2} + 8 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 6 x^{2} + 8 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 6 x^{2} + 8 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 6 x^{2} d x + \int 8 x d x$

Step 2.3.2

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

$y + C_{1} = 6 \int x^{2} d x + \int 8 x d x$

Step 2.3.3

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

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + \int 8 x d x$

Step 2.3.4

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

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 8 \int x d x$

Step 2.3.5

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

$y + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2}) + 8 (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Simplify.

$y + C_{1} = 6 (\frac{1}{3}) x^{3} + 8 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Combine $6$ and $\frac{1}{3}$ .

$y + C_{1} = \frac{6}{3} x^{3} + 8 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.6.2.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.6.2.2.1

Factor $3$ out of $6$ .

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

Step 2.3.6.2.2.2

Cancel the common factors.

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Step 2.3.6.2.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.6.2.2.2.2

Cancel the common factor.

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

Step 2.3.6.2.2.2.3

Rewrite the expression.

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

Step 2.3.6.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.6.2.3

Combine $8$ and $\frac{1}{2}$ .

$y + C_{1} = 2 x^{3} + \frac{8}{2} x^{2} + C_{4}$

Step 2.3.6.2.4

Cancel the common factor of $8$ and $2$ .

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Step 2.3.6.2.4.1

Factor $2$ out of $8$ .

$y + C_{1} = 2 x^{3} + \frac{2 \cdot 4}{2} x^{2} + C_{4}$

Step 2.3.6.2.4.2

Cancel the common factors.

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Step 2.3.6.2.4.2.1

Factor $2$ out of $2$ .

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

Step 2.3.6.2.4.2.2

Cancel the common factor.

$y + C_{1} = 2 x^{3} + \frac{2 \cdot 4}{2 \cdot 1} x^{2} + C_{4}$

Step 2.3.6.2.4.2.3

Rewrite the expression.

$y + C_{1} = 2 x^{3} + \frac{4}{1} x^{2} + C_{4}$

Step 2.3.6.2.4.2.4

Divide $4$ by $1$ .

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $3$ for $x$ and $11$ for $y$ in $y = 2 x^{3} + 4 x^{2} + K$ .

$11 = 2 \cdot 3^{3} + 4 \cdot 3^{2} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $2 \cdot 3^{3} + 4 \cdot 3^{2} + K = 11$ .

$2 \cdot 3^{3} + 4 \cdot 3^{2} + K = 11$

Step 4.2

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

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$2 \cdot 27 + 4 \cdot 3^{2} + K = 11$

Step 4.2.1.2

Multiply $2$ by $27$ .

$54 + 4 \cdot 3^{2} + K = 11$

Step 4.2.1.3

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

$54 + 4 \cdot 9 + K = 11$

Step 4.2.1.4

Multiply $4$ by $9$ .

$54 + 36 + K = 11$

Step 4.2.2

Add $54$ and $36$ .

$90 + K = 11$

Step 4.3

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

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Step 4.3.1

Subtract $90$ from both sides of the equation.

$K = 11 - 90$

Step 4.3.2

Subtract $90$ from $11$ .

$K = - 79$

Step 5

Substitute $- 79$ for $K$ in $y = 2 x^{3} + 4 x^{2} + K$ and simplify.

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Step 5.1

Substitute $- 79$ for $K$ .

$y = 2 x^{3} + 4 x^{2} - 79$