integration by parts

integral of x(x+1)^5 dx

Question

integration by parts 

integral of x(x+1)^5 dx

anonymous · Answer

$$\int\limits x(x+1)^5$$

kainui · Answer

I think you might be better off by using a substitution, such as x+1=u.

vishweshshrimali5 · Answer

@givanna618 , @Kainui is perfectly correct

vishweshshrimali5 · Answer

Though even substituting u = x will work

vishweshshrimali5 · Answer

But why bother using u then ?

tkhunny · Answer

You can always expand it.

$(x+1)^{5} = x^{5} + 5x^{4} + 10x^{3} + 10x^{2} + 5x + 1$

kainui · Answer

Yeah but that's really not necessary...

vishweshshrimali5 · Answer

$$\large{\int x(x+1)^5 \ dx}$$

$$\large{= \int (u-1)u^5 du}$$

$$\large{= \int u^6 du - \int u^5 du}$$

That is why @kainui suggested what he suggested

vishweshshrimali5 · Answer

Using, u = x+1, ensures that you don't have to use binomial theorem :)

vishweshshrimali5 · Answer

It gets easier that way

vishweshshrimali5 · Answer

Great work @Kainui :D

tkhunny · Answer

Necessary?  It's better than a brain cramp.  It's absolutely the hardest way to do it, but it is a way to do it.  If the alternative is to punt, why not do it a hard way?

vishweshshrimali5 · Answer

Very wise words @tkhunny :)

vishweshshrimali5 · Answer

Well @givanna618 now you have 2 different methods of solving the question thanks to @tkhunny and @Kainui . You can use whichever you like

vishweshshrimali5 · Answer

And you can also use integration by parts

vishweshshrimali5 · Answer

Though I would not prefer it ^^^

tkhunny · Answer

"by parts" was my third option.

vishweshshrimali5 · Answer

:) By parts is a very general method. It can be used in many many integrals.

vishweshshrimali5 · Answer

Sometimes I really wish there was an option of giving more than 1 medal.