Question

Indefinite Integral

Find the value of \[\large{ I = \int \cfrac{1}{x(x^n + 1)} dx }\]

Answer 1

@vishweshshrimali5

This is what I have tried yet :

\(\cfrac{1}{x(x^n +1) } = \cfrac{A}{x} + \cfrac{B}{x^n +1}\)

\(\cfrac{A(x^n) + A + Bx}{x(x^n + 1)}\)

Comparing this with the numerator of \(\cfrac{1}{x(x^n+1)}\) , we get :

A = 0 and B = 1

Answer 2

So, it can be written as :

\(\int \left( \cfrac{1}{x(x^n+1)} \right) = \int \left( \cfrac{1}{x^n + 1} \right) \)

Answer 3

Nope since, n is not given here you can not use such partial fractions.

Answer 4

Oh.. My bad. So, that all above is wrong? What to do now?

Answer 5

One way is to use integration by parts but I will not prefer it.

Answer 6

I am going to put x as e^u.

Answer 7

Then I get,

\[\large{\int\cfrac{e^u * du}{e^u(e^{nu} + 1)}}\]

Answer 8

OKay I think I got it

Answer 9

Just forget all this.

Answer 10

Multiply and divide by x^{n-1}

Answer 11

\[I=\int\limits \frac{ x ^{n-1}dx }{ x^n \left( x^n+1 \right) }\]
put
\[x^n=t,nx ^{n-1}dx=dt,\]

Answer 12

Okay, so, that becomes :

\[I = \int \frac{\frac{dt}{n}}{t(t+1)}\]

Answer 13

\[ I = \int \cfrac{dt}{n \times t(t+1)}\]

@Zarkon can you guide how to go on from here?

Answer 14

Should I use partial fractions now?

Answer 15

\[ I = \cfrac{1}{n} \int \cfrac{dt}{t(t+1)}\]

Answer 16

Okay. Got it, Thanks @vishweshshrimali5 and @surjithayer :-)

Using partial fraction will be the best I think.

Answer 17

you are correct partial fractions is the best option.