Calculus Examples

$a' (x) = \frac{2 x^{3} - 200}{x^{2}}$

Step 1

The function $a (x)$ can be found by evaluating the indefinite integral of the derivative $a' (x)$ .

$a (x) = \int a' (x) d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int (2 x^{3} - 200) {(x^{2})}^{- 1} d x$

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (2 x^{3} - 200) x^{2 \cdot - 1} d x$

Step 2.2.2

Multiply $2$ by $- 1$ .

$\int (2 x^{3} - 200) x^{- 2} d x$

Step 3

Multiply $(2 x^{3} - 200) x^{- 2}$ .

$\int 2 x^{3} x^{- 2} - 200 x^{- 2} d x$

Step 4

Simplify.

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Step 4.1

Multiply $x^{3}$ by $x^{- 2}$ by adding the exponents.

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Step 4.1.1

Move $x^{- 2}$ .

$\int 2 (x^{- 2} x^{3}) - 200 x^{- 2} d x$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 2 x^{- 2 + 3} - 200 x^{- 2} d x$

Step 4.1.3

Add $- 2$ and $3$ .

$\int 2 x^{1} - 200 x^{- 2} d x$

Step 4.2

Simplify $2 x^{1}$ .

$\int 2 x - 200 x^{- 2} d x$

Step 5

Split the single integral into multiple integrals.

$\int 2 x d x + \int - 200 x^{- 2} d x$

Step 6

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

$2 \int x d x + \int - 200 x^{- 2} d x$

Step 7

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

$2 (\frac{1}{2} x^{2} + C) + \int - 200 x^{- 2} d x$

Step 8

Since $- 200$ is constant with respect to $x$ , move $- 200$ out of the integral.

$2 (\frac{1}{2} x^{2} + C) - 200 \int x^{- 2} d x$

Step 9

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

$2 (\frac{1}{2} x^{2} + C) - 200 (- x^{- 1} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$2 (\frac{1}{2}) x^{2} - 200 (- x^{- 1}) + C$

Step 10.2

Simplify.

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Step 10.2.1

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

$\frac{2}{2} x^{2} - 200 (- x^{- 1}) + C$

Step 10.2.2

Cancel the common factor of $2$ .

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Step 10.2.2.1

Cancel the common factor.

$\frac{2}{2} x^{2} - 200 (- x^{- 1}) + C$

Step 10.2.2.2

Rewrite the expression.

$1 x^{2} - 200 (- x^{- 1}) + C$

Step 10.2.3

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

$x^{2} - 200 (- x^{- 1}) + C$

Step 10.2.4

Multiply $- 1$ by $- 200$ .

$x^{2} + 200 x^{- 1} + C$

Step 11

The function $a$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$a (x) = x^{2} + 200 x^{- 1} + C$