Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 37

int _(pi/4)^(pi/2) cot^5(x)csc^3(x)dx
To solve, apply the Pythagorean identity 1+cot^2(x)=csc^2(x) to express the integral in the form int u^n du.
=int _(pi/4)^(pi/2) cot^2 (x) cot^3(x) csc^3(x)dx
=int_(pi/4)^(pi/2) (csc^2(x)-1) cot^3(x)csc^3(x)dx
=int_(pi/4)^(pi/2)( csc^5(x)cot^3(x)-csc^3(x)cot^3(x) )dx
=int_(pi/4)^(pi/2) ( csc^5(x) cot(x) cot^2(x) - csc^3(x) cot(x)cot^2(x) )dx
=int_(pi/4)^(pi/2) ( csc^5(x) cot(x) (csc^2(x)-1) - csc^3(x) cot(x) (csc^2(x) -1)) dx
=int_(pi/4)^(pi/2) (csc^7(x)cot(x)-csc^5(x)cot(x)-csc^5(x)cot(x) +csc^3(x)cot(x)) dx
=int_(pi/4)^(pi/2) (csc^7(x)cot(x) -2csc^5(x)cot(x) +csc^3(x)cotx) dx
To take the integral, apply u-substitution method. So, let u be:
u=csc(x) dx
Then, differentiate u.
du = - csc(x) cot(x) dx
To be able to apply this, factor out -csc(x) cot(x).
=int_(pi/4)^(pi/2) ( -csc^6(x) +2csc^4(x) -csc^2(x)) (-csc(x) cot(x) dx)
Then, determine the value of u when x=pi/2 and x=pi/4.
u=csc (x)
u= csc(pi/2)=1
u=csc(pi/4)=sqrt2
Expressing the integral in terms of u, the integral becomes:
=int _sqrt2^1 (-u^6 +2u^4-u^2) du
=( -u^7/7 +(2u^5)/5 -u^3/3 )_sqrt2^1
=(-1/7+2/5-1/3) -(-(sqrt2)^7/7+(2(sqrt2)^5)/5-(sqrt2)^3/3)
=0.2201

Therefore, int _(pi/4)^(pi/2) cot^5(x) csc^3(x) dx=0.2201 .

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