Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Multiply $- 1$ by $3$ .

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