Trouble solving for anti-derivative: int{1/(x^2(x^6+1)^(5/6))}dx

[frank789]

Hi all

Having trouble with this integral: (let int denote integral)

int{1/(x^2(x^6+1)^(5/6))}dx

Heres what I have attempted/thought so far.

1. shouldnt make a substitution for (x^6+1) because it would leave me with an uncancelled factor of x

2. cant integrate by parts because I would either have to integrate (x^6+1)^(5/6) which would just keep increasing the power or differentiate it which would just keep decreasing it and what i feel like the crux of the problem is doesnt change.

3. then i thought id try to make u^(6/5) = x^6+1 to cancel the (-5/6) exponent but that would increase the power of x (at least according to what I did, rearranged dx = (u^(1/5)du/5x^5)

4. tried looking through my notes and the profs notes he posted online but I cant find a similar integration senario to go off of

5. tried simplifying the denominator by multiplying by (x^6+1)^(1/5)/((x^6+1)^(1/5)) but im still left powers of x that i cant see a substitution for

6. checked online integral solver for a hint. they said to let u = ((x^6+1)^1/6)/(x) which I can see right away is just one less power of each function. I worked through that and it in fact does work and when i sub back in eliminates the integral down to int{1}du.

I feel like there has to be a better strategyfor this one that im not seeing. Or is reducing powers a go to strategy for 1/some function integrals

If you could perhaps suggest a different starting point or offer some pointers on how to approach a problem like this I would appreciate it!

 

[Deleted member 4993]

Hi all

Having trouble with this integral: (let int denote integral)

int{1/(x^2(x^6+1)^(5/6))}dx

Heres what I have attempted/thought so far.

1. shouldnt make a substitution for (x^6+1) because it would leave me with an uncancelled factor of x

2. cant integrate by parts because I would either have to integrate (x^6+1)^(5/6) which would just keep increasing the power or differentiate it which would just keep decreasing it and what i feel like the crux of the problem is doesnt change.

3. then i thought id try to make u^(6/5) = x^6+1 to cancel the (-5/6) exponent but that would increase the power of x (at least according to what I did, rearranged dx = (u^(1/5)du/5x^5)

4. tried looking through my notes and the profs notes he posted online but I cant find a similar integration senario to go off of

5. tried simplifying the denominator by multiplying by (x^6+1)^(1/5)/((x^6+1)^(1/5)) but im still left powers of x that i cant see a substitution for

6. checked online integral solver for a hint. they said to let u = ((x^6+1)^1/6)/(x) which I can see right away is just one less power of each function. I worked through that and it in fact does work and when i sub back in eliminates the integral down to int{1}du.

I feel like there has to be a better strategyfor this one that im not seeing. Or is reducing powers a go to strategy for 1/some function integrals

If you could perhaps suggest a different starting point or offer some pointers on how to approach a problem like this I would appreciate it!

This is the time to call uncle "wolfram" (which I did) - and according to my uncle:

substitute u = 1/x⁶

and things fall in line.....

 

[frank789]

This is the time to call uncle "wolfram" (which I did) - and according to my uncle:

substitute u = 1/x⁶

and things fall in line.....

that it does haha. solved with the substitution (albeit modified). I solved for -(1+1/x^6)^(1/6).

What I took away from this problem, and applied to a later one with success was to factor out an x in the inside of the sixth root. when i factored out x^6 i brought it out from under the root sign and then i was easily able to make a substituition. the only difference is i substituted 1+1/x^6 as it made the integral cleaner and more obvious to me.

thanks for the help and sorry for the late reply!

 

[frank789]

+C of course