Calculus Examples

$x^{2.2} + x^{- 1.6}$

Step 1

Write $x^{2.2} + x^{- 1.6}$ as a function.

$f (x) = x^{2.2} + x^{- 1.6}$

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.

$F (x) = \int x^{2.2} + x^{- 1.6} d x$

Step 4

Split the single integral into multiple integrals.

$\int x^{2.2} d x + \int x^{- 1.6} d x$

Step 5

By the Power Rule, the integral of $x^{2.2}$ with respect to $x$ is $\frac{1}{3.2} x^{3.2}$ .

$\frac{1}{3.2} x^{3.2} + C + \int x^{- 1.6} d x$

Step 6

By the Power Rule, the integral of $x^{- 1.6}$ with respect to $x$ is $\frac{1}{- 0.6} x^{- 0.6}$ .

$\frac{1}{3.2} x^{3.2} + C + \frac{1}{- 0.6} x^{- 0.6} + C$

Step 7

Simplify.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{3.2} x^{3.2} + C - \frac{1}{0.6} x^{- 0.6} + C$

Step 7.2

Simplify.

$0.3125 x^{3.2} - \frac{1}{0.6} x^{- 0.6} + C$

Step 8

The answer is the antiderivative of the function $f (x) = x^{2.2} + x^{- 1.6}$ .

$F (x) =$ $0.3125 x^{3.2} - \frac{1}{0.6} x^{- 0.6} + C$