Antiderivative, if x is in the denominator

"x" in the denominator - this is how you crack the integral

• For the integral of a power function f (x) = xⁿ have you developed a formula? got to know. The antiderivative F (x) = 1 / n + 1 applies _* xⁿ+1. With this formula you can find the antiderivatives of all power functions, but also of completely rational ones Functions to calculate.
• As with the derivation, this formula has a huge advantage, because it does not only apply to natural ones Counting as an exponent, but also when the exponent is a whole, a rational or even a real number, with the exception of f (x) = 1 / x - a special case (see below).
• Accordingly, it is possible to integrate functions in which the unknown "x" occurs as a power in the denominator using this formula. All you have to do is write the function as a negative power using the power laws.
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• For f (x) = 1 / x² = x^-2 you get (insert n = -2 in the formula!) accordingly F (x) = 1 / -1 _* x^-1 = -1 / x. Even f (x) = 1 / √x = x^-1/2 you can integrate accordingly (n = -1/2) and get F (x) = 2 _* x¹/2 = 2 _* √x.

The special case 1 / x and other pitfalls with the antiderivative

• The function f (x) = 1 / x = x^-1 is a special case, because if you insert n = -1 in the formula for the antiderivative, then the denominator of the coefficient 1 / n + 1 becomes zero. In fact, this integral cannot be solved with the simple formula. The antiderivative is F (x) = ln x, the natural logarithm - you just have to remember this exception.


Derive 2 by x - this is how it works with fractional-rational functions

If you want to derive the function "2 by x", you can do this with a little ...

• Compound functions, in which "x" appears in the denominator, are of course more complicated and can no longer be cracked with a simple formula. For example, to integrate f (x) = x / (x² -1) or f (x) = e^x/ x further integration rules (tip: integration boards on the Internet and in many formulas are helpful). And some functions cannot be integrated at all, in other words: The antiderivative F (x) cannot be specified in a closed form.