Special Case Anti-Derivative

[Jason76]

With intergals x to the 0 power cannot work with the intergal power rule. Therefore, the part which comes out to a 0 exponent (after applying the power rule) becomes an $\displaystyle ln$.

$\displaystyle \int 5x^{-1} + x^{4} + x$

$\displaystyle 5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

or

$\displaystyle -5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

Which one?

 

[mmm4444bot]

Neither is correct. Check your antiderivative for x^4.

Your may check your answer by differentiation; the derivative of your answer must be the integrand, yes?

Where is the dx, in your original post?

Also, I'm not sure why you're discussing x^0.

x^0 = 1

 

[lookagain]

With intergals x to the 0 power cannot work with the intergal power rule. Therefore, the part which comes out to a 0 exponent (after applying the power rule) becomes an $\displaystyle ln$.

$\displaystyle \int (5x^{-1} + x^{4} + x)dx$

. . . This is one of the versions of a corrected form of the integral.


$\displaystyle 5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

. . . No, the antiderivative of 1/x is ln|x| + C, not |ln x| + C.


(And it was already pointed out that the antiderivative of the second term
is not what you typed. The same is true for the other expression below.)

or

$\displaystyle -5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

Which one?

"With integral x to the 0 power cannot work with the integral power rule.
Therefore, the part which comes out to a 0 exponent (after applying the power rule) becomes an [FONT=ea9bd3dac1f0b2790750af38#e40700][FONT=MathJax_Math-italic]l[/FONT][FONT=MathJax_Math-italic]n[/FONT]

[/FONT] ."


You must have meant that the antiderivative of x to the -1 power cannot work the integral power rule.


Note the spelling of "integral."

 

[Jason76]

Let's look at just the 5x

$\displaystyle \int 5x^{-1} dx$

Comes out to

$\displaystyle \frac{-5x^{0}}{0} + C$ or $\displaystyle \frac{-5}{0} + C$

once the integration power rule is applied. Since division by 0 is undefined, our answer has to be some $\displaystyle ln$ form.

Let's look at the other part by itself:

$\displaystyle \int x^{4} dx$

and so came out with

$\displaystyle \frac{4x^{5}}{5} + C$

integration power rule:

$\displaystyle \frac{nx^{n + 1}}{n + 1} + C$

 

[JeffM]

With intergals x to the 0 power cannot work with the intergal power rule. Therefore, the part which comes out to a 0 exponent (after applying the power rule) becomes an $\displaystyle ln$.

$\displaystyle \int 5x^{-1} + x^{4} + x$

$\displaystyle 5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

or

$\displaystyle -5|ln x| + \frac{4x^{5}}{5} + \frac{x^{2}}{2} + C$

Which one?

The rule is $\displaystyle x \ne 0 \implies \int \dfrac{1}{x}dx = ln(|x|) + C.$