Calculus Examples

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

Step 1

Multiply $2 x (1 - x^{- 3})$ .

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

Step 2

Simplify.

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Step 2.1

Multiply $2$ by $1$ .

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

Step 2.2

Multiply $- 1$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

Move $x^{- 3}$ .

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

Step 2.3.2

Multiply $x^{- 3}$ by $x$ .

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Step 2.3.2.1

Raise $x$ to the power of $1$ .

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

Step 2.3.2.2

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

$\int 2 x - 2 x^{- 3 + 1} d x$

Step 2.3.3

Add $- 3$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Cancel the common factor.

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

Step 8.2.2.2

Rewrite the expression.

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

Step 8.2.3

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

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

Step 8.2.4

Multiply $- 1$ by $- 2$ .

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