Integration of these 3 questions

With full explanation and all steps and the formulaes used..

himani only ask 1 question per thread, as it is not possible to ans so many of ur ques.

ques 1: substitute y= x^3, integral will become smth like y/(y^3 – 1). write the denominator y^3 – 1 as (y – 1)(y^2+y+1) and then use partial fractions and standard methods to integrate terms of the forms L/Q (linear/quad).

ques 2: 1/x^13(x^6 – 1)= x^5/x^18(x^6 – 1). now substitute x^6= y so it will become smth like 1/y^3(y – 1) now use partial fracs.

ques 3: this is bit tricky one. divide the numerator and denominator by x^10, you ll get (x^4 – x^-6)/(x^10+x^-10 – 3). now aubstitute y= x^5 + x^-5 and note that y^2 – 2= x^10+x^-10. it then becomes smth like 1/(y^2 – 5)= 1/(y – root5)(y+root5). use partial fracs now.

note that if u don know abt partial fractions then you ll have a hard time solving these. so, you need to understand them first (they are very ez btw) and then come back to these integrals.