Calculus Examples

$\frac{d y}{d x} = x^{4} - x^{3} + x - 1$ , $f (0) = - 3$

Step 1

Rewrite the equation.

$d y = (x^{4} - x^{3} + x - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x^{4} - x^{3} + x - 1 d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Apply the constant rule.

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

Step 2.3.8

Simplify.

$y + C_{1} = \frac{1}{5} x^{5} - \frac{1}{4} x^{4} + \frac{1}{2} x^{2} - x + C_{6}$

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $- 3$ for $y$ in $y = \frac{1}{5} x^{5} - \frac{1}{4} x^{4} + \frac{1}{2} x^{2} - x + K$ .

$- 3 = \frac{1}{5} \cdot 0^{5} - \frac{1}{4} \cdot 0^{4} + \frac{1}{2} \cdot 0^{2} - 0 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{1}{5} \cdot 0^{5} - \frac{1}{4} \cdot 0^{4} + \frac{1}{2} \cdot 0^{2} - 0 + K = - 3$ .

$\frac{1}{5} \cdot 0^{5} - \frac{1}{4} \cdot 0^{4} + \frac{1}{2} \cdot 0^{2} - 0 + K = - 3$

Step 4.2

Simplify $\frac{1}{5} \cdot 0^{5} - \frac{1}{4} \cdot 0^{4} + \frac{1}{2} \cdot 0^{2} - 0 + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{5} \cdot 0 - \frac{1}{4} \cdot 0^{4} + \frac{1}{2} \cdot 0^{2} - 0 + K = - 3$

Step 4.2.1.2

Multiply $\frac{1}{5}$ by $0$ .

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

Step 4.2.1.3

Raising $0$ to any positive power yields $0$ .

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

Step 4.2.1.4

Multiply $- \frac{1}{4} \cdot 0$ .

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Step 4.2.1.4.1

Multiply $0$ by $- 1$ .

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

Step 4.2.1.4.2

Multiply $0$ by $\frac{1}{4}$ .

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

Step 4.2.1.5

Raising $0$ to any positive power yields $0$ .

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

Step 4.2.1.6

Multiply $\frac{1}{2}$ by $0$ .

$0 + 0 + 0 - 0 + K = - 3$

Step 4.2.1.7

Multiply $- 1$ by $0$ .

$0 + 0 + 0 + 0 + K = - 3$

Step 4.2.2

Combine the opposite terms in $0 + 0 + 0 + 0 + K$ .

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Step 4.2.2.1

Add $0$ and $0$ .

$0 + 0 + 0 + K = - 3$

Step 4.2.2.2

Add $0$ and $0$ .

$0 + 0 + K = - 3$

Step 4.2.2.3

Add $0$ and $0$ .

$0 + K = - 3$

Step 4.2.2.4

Add $0$ and $K$ .

$K = - 3$

Step 5

Substitute $- 3$ for $K$ in $y = \frac{1}{5} x^{5} - \frac{1}{4} x^{4} + \frac{1}{2} x^{2} - x + K$ and simplify.

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Step 5.1

Substitute $- 3$ for $K$ .

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

Step 5.2

Simplify each term.

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Step 5.2.1

Combine $\frac{1}{5}$ and $x^{5}$ .

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

Step 5.2.2

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

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

Step 5.2.3

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

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