Calculus Examples

$\frac{d y}{d x} = x^{2} - 1$ , $y (3) = 7$

Step 1

Rewrite the equation.

$d y = (x^{2} - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $3$ for $x$ and $7$ for $y$ in $y = \frac{1}{3} x^{3} - x + K$ .

$7 = \frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + K = 7$ .

$\frac{1}{3} \cdot 3^{3} - 1 \cdot 3 + K = 7$

Step 4.2

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

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $3^{3}$ .

$\frac{1}{3} \cdot (3 \cdot 3^{2}) - 1 \cdot 3 + K = 7$

Step 4.2.1.1.2

Cancel the common factor.

$\frac{1}{3} \cdot (3 \cdot 3^{2}) - 1 \cdot 3 + K = 7$

Step 4.2.1.1.3

Rewrite the expression.

$3^{2} - 1 \cdot 3 + K = 7$

Step 4.2.1.2

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

$9 - 1 \cdot 3 + K = 7$

Step 4.2.1.3

Multiply $- 1$ by $3$ .

$9 - 3 + K = 7$

Step 4.2.2

Subtract $3$ from $9$ .

$6 + K = 7$

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 $6$ from both sides of the equation.

$K = 7 - 6$

Step 4.3.2

Subtract $6$ from $7$ .

$K = 1$

Step 5

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

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Step 5.1

Substitute $1$ for $K$ .

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

Step 5.2

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

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