Calculus Examples

$\frac{d y}{d x} = x^{3} - x^{2} + 2 x$ , $y (2) = 0$

Step 1

Rewrite the equation.

$d y = (x^{3} - x^{2} + 2 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{3} - x^{2} + 2 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x^{3} - x^{2} + 2 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int x^{3} d x + \int - x^{2} d x + \int 2 x d x$

Step 2.3.2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 2.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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

Step 2.3.7.2

Simplify.

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Step 2.3.7.2.1

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

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

Step 2.3.7.2.2

Cancel the common factor of $2$ .

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Step 2.3.7.2.2.1

Cancel the common factor.

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

Step 2.3.7.2.2.2

Rewrite the expression.

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

Step 2.3.7.2.3

Multiply $x^{2}$ by $1$ .

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

Step 2.3.8

Reorder terms.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $2$ for $x$ and $0$ for $y$ in $y = x^{2} + \frac{1}{4} x^{4} - \frac{1}{3} x^{3} + K$ .

$0 = 2^{2} + \frac{1}{4} \cdot 2^{4} - \frac{1}{3} \cdot 2^{3} + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $2^{2} + \frac{1}{4} \cdot 2^{4} - \frac{1}{3} \cdot 2^{3} + K = 0$ .

$2^{2} + \frac{1}{4} \cdot 2^{4} - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2

Simplify $2^{2} + \frac{1}{4} \cdot 2^{4} - \frac{1}{3} \cdot 2^{3} + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$4 + \frac{1}{4} \cdot 2^{4} - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2.1.2

Raise $2$ to the power of $4$ .

$4 + \frac{1}{4} \cdot 16 - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2.1.3

Cancel the common factor of $4$ .

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Step 4.2.1.3.1

Factor $4$ out of $16$ .

$4 + \frac{1}{4} \cdot (4 (4)) - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2.1.3.2

Cancel the common factor.

$4 + \frac{1}{4} \cdot (4 \cdot 4) - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2.1.3.3

Rewrite the expression.

$4 + 4 - \frac{1}{3} \cdot 2^{3} + K = 0$

Step 4.2.1.4

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

$4 + 4 - \frac{1}{3} \cdot 8 + K = 0$

Step 4.2.1.5

Multiply $- \frac{1}{3} \cdot 8$ .

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Step 4.2.1.5.1

Multiply $8$ by $- 1$ .

$4 + 4 - 8 (\frac{1}{3}) + K = 0$

Step 4.2.1.5.2

Combine $- 8$ and $\frac{1}{3}$ .

$4 + 4 + \frac{- 8}{3} + K = 0$

Step 4.2.1.6

Move the negative in front of the fraction.

$4 + 4 - \frac{8}{3} + K = 0$

Step 4.2.2

Add $4$ and $4$ .

$8 - \frac{8}{3} + K = 0$

Step 4.2.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} - \frac{8}{3} + K = 0$

Step 4.2.4

Combine $8$ and $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3} - \frac{8}{3} + K = 0$

Step 4.2.5

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 - 8}{3} + K = 0$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $8$ by $3$ .

$\frac{24 - 8}{3} + K = 0$

Step 4.2.6.2

Subtract $8$ from $24$ .

$\frac{16}{3} + K = 0$

Step 4.3

Subtract $\frac{16}{3}$ from both sides of the equation.

$K = - \frac{16}{3}$

Step 5

Substitute $- \frac{16}{3}$ for $K$ in $y = x^{2} + \frac{1}{4} x^{4} - \frac{1}{3} x^{3} + K$ and simplify.

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Step 5.1

Substitute $- \frac{16}{3}$ for $K$ .

$y = x^{2} + \frac{1}{4} x^{4} - \frac{1}{3} x^{3} - \frac{16}{3}$

Step 5.2

Simplify each term.

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Step 5.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$y = x^{2} + \frac{x^{4}}{4} - \frac{1}{3} x^{3} - \frac{16}{3}$

Step 5.2.2

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

$y = x^{2} + \frac{x^{4}}{4} - \frac{x^{3}}{3} - \frac{16}{3}$