integrate ∫(1/(x(x^6-16)^1/2))dx plssolve it and send the solution

∫1/(x(x^6-16)^1/2)) dx     Multiply and divide by x^2  u get  ∫x^2 / x^3((x^6-16)^1/2)     Put x^3=t     dt/3=x^2dx   So now ur integration is

∫1/t((t^2-16)^1/2))dt    Take the t outside the root inside u get  ∫1/((t^4-16t^2)^1/2))   Now complete the square and u will get        ∫1/(((t^2-8)^2-8^2)^1/2) now using general formula                                                  ∫1/(x^2-a^2)^1/2 dx = log(x+(x^2-a^2)^1-2)            Here x is t^2-8 and a is 8    So substitute and substitute back value of x^3 which is t.  Ur final ans will be                                    log(x^6-8+(((x^6-8)^2)-8^2)^1/2)

 

 

 

 

 

 

Hello Vinod! did you mean the following integral?

For this integral there exists no closed function as an answer- i.e., the integral isn’t elementary function!

This is actually out of class 12 syllabus I think.

We can answer it if it has limits. Many indefinite integrals donot have closed answers.

 

Let T = ∫(1/x dx = ∫x² / x³ 

Let x³ = 1/t   3x²dx = – 1/t²dt  i.e. x²dx = – 1/3t²dt 

So, T = ∫[(–1/3t²)/{(1 / t)}] dt = (–1/3)∫ 1 /[ t / t]

         = (–1/3)∫ [1 /] dt = (1/3)∫ [–1 /] dt

T = (1/3)(1/4)cos^–1(4t) + c  =  (1/12)cos^–1(4t) + c