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C4 Integration

integrate 1/(sin^2 x.cos^2 x)

please help guys
Reply 1
Avatar for Jonny W
Jonny W
8
Using the identity sin(2x) = 2sin(x)cos(x),

1 / (sin^2(x)cos^2(x))
= 1 / ((1/2)sin(2x))^2
= 4cosec^2(2x)

(int) 1 / (sin^2(x)cos^2(x)) dx
= (int) 4cosec^2(2x) dx
= (int) 2cosec^2(u) du . . . . . u = 2x, du/dx = 2
= -2cot(u) + c . . . . . using the formula booklet
= -2cot(2x) + c
Reply 2
Avatar for melbourne
melbourne
OP
Jonny W
1 / (sin^2(x)cos^2(x))
= 1 / ((1/2)sin(2x))^2
= 4cosec^2(2x)

(int) 1 / (sin^2(x)cos^2(x)) dx
= (int) 4cosec^2(2x) dx
= (int) 2cosec^2(u) du . . . . . u = 2x, du/dx = 2
= -2cot(u) + c . . . . . using the formula booklet
= -2cot(2x) + c


in the second method i see you used the substitution method, what did you do in the first example exactly?

thanks!
Reply 3
Avatar for dvs
dvs
10
I = ∫ 1/(sinxcosx)² dx
= ∫ 1/(0.5sin(2x))² dx
= ∫ (2cosec(2x))² dx
= ∫ 4cosec²(2x) dx
= -4 ∫ -cosec²(2x) dx
= -4 (1/2) cot(2x) + C
= -2 cot(2x) + C

Or:
I = ∫ (sin²x + cos²x)/(sinxcosx)²
= ∫ (sin²x)/(sinxcosx)² dx + ∫ (cos²x)/(sinxcosx)²
= ∫ 1/cos²x dx + ∫ 1/sin²x dx
= ∫ sec²x dx + ∫ cosec²x dx
= tanx - cotx + C
Reply 4
Avatar for melbourne
melbourne
OP
dvs
I = ∫ 1/(sinxcosx)² dx
= ∫ 1/(0.5sin(2x))² dx
= ∫ (2cosec(2x))² dx
= ∫ 4cosec²(2x) dx
= -4 ∫ -cosec²(2x) dx
= -4 (1/2) cot(2x) + C
= -2 cot(2x) + C

Or:
I = ∫ (sin²x + cos²x)/(sinxcosx)²
= ∫ (sin²x)/(sinxcosx)² dx + ∫ (cos²x)/(sinxcosx)²
= ∫ 1/cos²x dx + ∫ 1/sin²x dx
= ∫ sec²x dx + ∫ cosec²x dx
= tanx - cotx + C


ah sorry guys- i see what you've done- used trig (additon formulae)!!!!!

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