Calculus Examples

$\int (2 - \frac{3}{x^{10}}) d x$

Step 1

Remove parentheses.

$\int 2 - \frac{3}{x^{10}} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 d x + \int - \frac{3}{x^{10}} d x$

Step 3

Apply the constant rule.

$2 x + C + \int - \frac{3}{x^{10}} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 x + C - \int \frac{3}{x^{10}} d x$

Step 5

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

$2 x + C - (3 \int \frac{1}{x^{10}} d x)$

Step 6

Simplify the expression.

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Step 6.1

Multiply $3$ by $- 1$ .

$2 x + C - 3 \int \frac{1}{x^{10}} d x$

Step 6.2

Move $x^{10}$ out of the denominator by raising it to the $- 1$ power.

$2 x + C - 3 \int {(x^{10})}^{- 1} d x$

Step 6.3

Multiply the exponents in ${(x^{10})}^{- 1}$ .

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Step 6.3.1

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

$2 x + C - 3 \int x^{10 \cdot - 1} d x$

Step 6.3.2

Multiply $10$ by $- 1$ .

$2 x + C - 3 \int x^{- 10} d x$

Step 7

By the Power Rule, the integral of $x^{- 10}$ with respect to $x$ is $- \frac{1}{9} x^{- 9}$ .

$2 x + C - 3 (- \frac{1}{9} x^{- 9} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $x^{- 9}$ and $\frac{1}{9}$ .

$2 x + C - 3 (- \frac{x^{- 9}}{9} + C)$

Step 8.1.2

Move $x^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x + C - 3 (- \frac{1}{9 x^{9}} + C)$

Step 8.2

Simplify.

$2 x - \frac{- 1}{3 x^{9}} + C$

Step 8.3

Simplify.

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Step 8.3.1

Move the negative in front of the fraction.

$2 x - - \frac{1}{3 x^{9}} + C$

Step 8.3.2

Multiply $- 1$ by $- 1$ .

$2 x + 1 \frac{1}{3 x^{9}} + C$

Step 8.3.3

Multiply $\frac{1}{3 x^{9}}$ by $1$ .

$2 x + \frac{1}{3 x^{9}} + C$