Calculus Examples

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

Write $\cos^{- 1} (x)$ as a function.

$f (x) = \cos^{- 1} (x)$

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

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

Set up the integral to solve.

$F (x) = \int \cos^{- 1} (x) d x$

Simplify.

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Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{1}{\cos (x)} d x$

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\int \sec (x) d x$

The integral of $\sec (x)$ with respect to $x$ is $\ln (| \sec (x) + \tan (x) |)$ .

$\ln (| \sec (x) + \tan (x) |) + C$

The answer is the antiderivative of the function $f (x) = \cos^{- 1} (x)$ .

$F (x) =$ $\ln (| \sec (x) + \tan (x) |) + C$