Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.2.2

Multiply $\cos (x)$ by $1$ .

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