Calculus II - Trigonometric Integrals

[GreenPrint]

Homework Statement

Evaluate
integral csc(x)^4/cot(x)^2 dx

Homework Equations

The Attempt at a Solution

Apparently I'm doing something wrong, what I'm not sure, thanks for any help

My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c

integral csc(x)^4/cot(x)^2 dx

used fact that csc(x) = 1/sin(x)
used fact that cot(x) = 1/tan(x)
used fact that tan(x) = sin(x)/cos(x)

integral sin(x)^2/(sin(x)^4*cos(x)^2) dx

used the fact that x^m/x^n = x^(m-n)
used the fact that x^-m = 1/x^m

integral dx/(sin(x)^2*cos(x)^2)

used the fact that sin(x)^2 + cos(x)^2 = 1
solved for sin(x)^2
sin(x)^2 = 1 - cos(x)^2

integral dx/((1-cos(x)^2)*cos(x)^2)

expanded

integral dx/(cos(x)^2 - cos(x)^4)

broke into two separate integrals

integral dx/cos(x)^2 - integral dx/cos(x)^4

used fact that 1/cos(x) = sec(x)

integral sec(x)^2 dx - integral sec(x)^4 dx

begun evaluating first integral using
integral sec(x)^n = ( sec(x)^(n-2)*tan(x) )/( n-1 ) + ( n-2)/(n-1)*integral sec(x)^(n-2) dx, n =/= 1

(sec(x)^(2-2)*tan(x))/(2-1) - (2-2)/(2-1) integral sec(x)^(2-2) dx - integral sec(x)^4 dx

simplified

tan(x) - integral sec(x)^4 dx

begun evaluating second integral

tan(x) - [(sec(x)^(4-2)*tan(x))/(4-1) - integral sec(x)^2 dx]

simplified

tan(x) - (sec(x)^2*tan(x))/3 + integral sec(x)^2 dx

established already that integral sec(x)^2 dx = tan(x)

2*tan(x) - (sec(x)^2*tan(x))/3 + c

Wolfram Alpha Answer: -2 cot(2 x)+constant

MATLAB Answer: tan(x) - 1/tan(x)

Back of the Book Answer: tan(x) - cot(x) + c

My answer is not equivalent
2*tan(5) - (sec(5)^2*tan(5))/3 is about 7.243183523
tan(5) - 1/tan(5) is about -3.084702091
-2 cot(2*5) is about -3.084702091
tan(5) - cot(5) is about -3.084702091

 

[Herr Malus]

The way you broke up the fraction is incorrect, you can't expand a minus sign in the denominator as you did. I suggest you start over, and this time instead expand the numerator with the identity csc²(x)=1+cot²(x). You will then find this problem to be significantly easier.

 

[SammyS]

Homework Statement

Evaluate
integral csc(x)^4/cot(x)^2 dx

Homework Equations

The Attempt at a Solution

Apparently I'm doing something wrong, what I'm not sure, thanks for any help

My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c

integral csc(x)^4/cot(x)^2 dx

used fact that csc(x) = 1/sin(x)
used fact that cot(x) = 1/tan(x)
used fact that tan(x) = sin(x)/cos(x)

integral sin(x)^2/(sin(x)^4*cos(x)^2) dx

used the fact that x^m/x^n = x^(m-n)
used the fact that x^-m = 1/x^m

integral dx/(sin(x)^2*cos(x)^2)
...

At this point use the double angle formula for sine: sin(2x) = 2sin(x)*cos(x).