Integrating different kinds of functions

**No Logic Sense** · November 29th 2008, 07:02 AM

I have an assignment, which is to show the proofs for

$f(x)=x^{n}$
$F(x)=\frac{x^{n+1}}{n+1}+k$

And also proofs for how you integrate logarithm functions:
$log(x), ln(x)$

And trigonometric functions:
$sin(x), cos(x), tan(x)$

I looked in my books, but they don't have any proofs for why you get what you get when integrating them.

**Krizalid** · November 29th 2008, 08:26 AM

Originally Posted by No Logic Sense

I have an assignment, which is to show the proofs for

$f(x)=x^{n}$
$F(x)=\frac{x^{n+1}}{n+1}+k$

$\int{x^{n}\,dx}=\int{e^{n\ln x}\,dx}.$ Now put $u=\ln x$ and the integral becomes $\int e^{(n+1)u}\,du=\frac{e^{(n+1)\ln x}}{n+1}+k=\frac{x^{n+1}}{n+1}+k.$

**No Logic Sense** · November 29th 2008, 09:16 AM

I'm sorry, but I don't quite understand how you got the natural constant (e) (I'm not sure if that is what it's called in English) or how you got the +1 later on.

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