Calculus Examples

$x^{2} + \frac{1}{x^{2}}$

Step 1

Write $x^{2} + \frac{1}{x^{2}}$ as a function.

$f (x) = x^{2} + \frac{1}{x^{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{2} + \frac{1}{x^{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{2} d x + \int \frac{1}{x^{2}} d x$

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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

Step 9

The answer is the antiderivative of the function $f (x) = x^{2} + \frac{1}{x^{2}}$ .

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