Calculus Examples

$\int (- 2 x + 3 x^{- 3} - 4) d x$

Step 1

Remove parentheses.

$\int - 2 x + 3 x^{- 3} - 4 d x$

Step 2

Split the single integral into multiple integrals.

$\int - 2 x d x + \int 3 x^{- 3} d x + \int - 4 d x$

Step 3

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

$- 2 \int x d x + \int 3 x^{- 3} d x + \int - 4 d x$

Step 4

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

$- 2 (\frac{1}{2} x^{2} + C) + \int 3 x^{- 3} d x + \int - 4 d x$

Step 5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$- 2 (\frac{1}{2} x^{2} + C) + 3 \int x^{- 3} d x + \int - 4 d x$

Step 6

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

$- 2 (\frac{1}{2} x^{2} + C) + 3 (- \frac{1}{2} x^{- 2} + C) + \int - 4 d x$

Step 7

Simplify.

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Step 7.1

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

$- 2 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{2} x^{- 2} + C) + \int - 4 d x$

Step 7.2

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

$- 2 (\frac{x^{2}}{2} + C) + 3 (- \frac{x^{- 2}}{2} + C) + \int - 4 d x$

Step 7.3

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{2 x^{2}} + C) + \int - 4 d x$

Step 8

Apply the constant rule.

$- 2 (\frac{x^{2}}{2} + C) + 3 (- \frac{1}{2 x^{2}} + C) - 4 x + C$

Step 9

Simplify.

$- x^{2} - \frac{3}{2 x^{2}} - 4 x + C$