Calculus Examples

$\frac{d y}{d x} = x^{3}$ , $y (- 1) = 2$

Step 1

Rewrite the equation.

$d y = x^{3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{3} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

Solve for $K$ .

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Step 4.1

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

Raise $- 1$ to the power of $4$ .

$\frac{1}{4} \cdot 1 + K = 2$

Step 4.2.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} + K = 2$

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 $\frac{1}{4}$ from both sides of the equation.

$K = 2 - \frac{1}{4}$

Step 4.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$K = 2 \cdot \frac{4}{4} - \frac{1}{4}$

Step 4.3.3

Combine $2$ and $\frac{4}{4}$ .

$K = \frac{2 \cdot 4}{4} - \frac{1}{4}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{2 \cdot 4 - 1}{4}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $2$ by $4$ .

$K = \frac{8 - 1}{4}$

Step 4.3.5.2

Subtract $1$ from $8$ .

$K = \frac{7}{4}$

Step 5

Substitute $\frac{7}{4}$ for $K$ in $y = \frac{1}{4} x^{4} + K$ and simplify.

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Step 5.1

Substitute $\frac{7}{4}$ for $K$ .

$y = \frac{1}{4} x^{4} + \frac{7}{4}$

Step 5.2

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

$y = \frac{x^{4}}{4} + \frac{7}{4}$