Calculus Examples

$\frac{d f}{d x} = 12 x^{5} - 4 x + 3$ , $f (- 1) = 3$

Step 1

Rewrite the equation.

$d f = (12 x^{5} - 4 x + 3) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d f = \int 12 x^{5} - 4 x + 3 d x$

Step 2.2

Apply the constant rule.

$f + C_{1} = \int 12 x^{5} - 4 x + 3 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$f + C_{1} = \int 12 x^{5} d x + \int - 4 x d x + \int 3 d x$

Step 2.3.2

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$f + C_{1} = 12 \int x^{5} d x + \int - 4 x d x + \int 3 d x$

Step 2.3.3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$f + C_{1} = 12 (\frac{1}{6} x^{6} + C_{2}) + \int - 4 x d x + \int 3 d x$

Step 2.3.4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$f + C_{1} = 12 (\frac{1}{6} x^{6} + C_{2}) - 4 \int x d x + \int 3 d x$

Step 2.3.5

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

$f + C_{1} = 12 (\frac{1}{6} x^{6} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + \int 3 d x$

Step 2.3.6

Apply the constant rule.

$f + C_{1} = 12 (\frac{1}{6} x^{6} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + 3 x + C_{4}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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Step 2.3.7.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$f + C_{1} = 12 (\frac{x^{6}}{6} + C_{2}) - 4 (\frac{1}{2} x^{2} + C_{3}) + 3 x + C_{4}$

Step 2.3.7.1.2

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

$f + C_{1} = 12 (\frac{x^{6}}{6} + C_{2}) - 4 (\frac{x^{2}}{2} + C_{3}) + 3 x + C_{4}$

Step 2.3.7.2

Simplify.

$f + C_{1} = 2 x^{6} - 2 x^{2} + 3 x + C_{5}$

Step 2.4

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

$f = 2 x^{6} - 2 x^{2} + 3 x + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $- 1$ for $x$ and $3$ for $f$ in $f = 2 x^{6} - 2 x^{2} + 3 x + K$ .

$3 = 2 {(- 1)}^{6} - 2 {(- 1)}^{2} + 3 \cdot - 1 + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $2 {(- 1)}^{6} - 2 {(- 1)}^{2} + 3 \cdot - 1 + K = 3$ .

$2 {(- 1)}^{6} - 2 {(- 1)}^{2} + 3 \cdot - 1 + K = 3$

Step 4.2

Simplify $2 {(- 1)}^{6} - 2 {(- 1)}^{2} + 3 \cdot - 1 + K$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 1$ to the power of $6$ .

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

Step 4.2.1.2

Multiply $2$ by $1$ .

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

Step 4.2.1.3

Raise $- 1$ to the power of $2$ .

$2 - 2 \cdot 1 + 3 \cdot - 1 + K = 3$

Step 4.2.1.4

Multiply $- 2$ by $1$ .

$2 - 2 + 3 \cdot - 1 + K = 3$

Step 4.2.1.5

Multiply $3$ by $- 1$ .

$2 - 2 - 3 + K = 3$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $2$ from $2$ .

$0 - 3 + K = 3$

Step 4.2.2.2

Subtract $3$ from $0$ .

$- 3 + K = 3$

Step 4.3

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

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Step 4.3.1

Add $3$ to both sides of the equation.

$K = 3 + 3$

Step 4.3.2

Add $3$ and $3$ .

$K = 6$

Step 5

Substitute $6$ for $K$ in $f = 2 x^{6} - 2 x^{2} + 3 x + K$ and simplify.

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Step 5.1

Substitute $6$ for $K$ .

$f = 2 x^{6} - 2 x^{2} + 3 x + 6$