prove that integral a to b ((bx)-1)^(0.5) equalto b[cos^-1(ab)_ ((ab)^(0.5))(1_ (ab))^(0.5)]

prove that integral a to b ((bx)-1)^(0.5) equalto b[cos^-1(ab)_ ((ab)^(0.5))(1_ (ab))^(0.5)]

Grade:12th Pass

1 Answers

Aman Bansal
592 Points
10 years ago

Dear Shakeel,

T: trigonometric functions: sin(x), tan(x), etc. 


Function Indefinite Integral Comments
sin x -cos x  
cos x sin x  
sec2 x tan x  
tan2 x = sec2x - 1 tan(x) - x  
(sec x)(tan x) = sin x / cos2 x sec x  
csc2 x -cot x  
cot2 x = csc2x - 1 -x - cot x  
(csc x)(cot x) = cos x / sin2 x -csc x  
tan x -ln cos x  
cot x = 1/(tan x) ln sin x  
sec x =
(sec x tan x + sec2 x)/(sec x + tan x) =
1/(cos^2(x/2) - sin^2(x/2)) = sec^2(x/2) / (1-tan^2(x/2))
ln(sec x + tan x), or
2 arctanh(tan(x/2)
csc x =
(-csc x cot x + csc2 x)/(csc x - cot x) =
= (1/2)(csc(x/2) sec(x/2)) = (1/2)sec^2(x/2)/tan(x/2)
ln(csc x - cot x), or
sin2 x =
(1/2)(x - sin x cos x) =
x/2 - sin(2x)/4
cos2 x = 1 - sin2x x/2 + sin(2x)/4  
sin(ax)sin(bx) (1/2)(sin(a-b)x/(a-b) - sin(a+b)x/(a+b)) a ≠ ±b, Proof
sin(ax)cos(bx) -(1/2)(cos(a-b)x/(a-b) + cos(a+b)x/(a+b)) a ≠ ±b, Proof
cos(ax)cos(bx) (1/2)(sin(a-b)x/(a-b) + sin(a+b)x/(a+b)) a ≠ ±b, Proof
x sin(ax) (sin(ax) - ax cos(ax))/a2  
x cos(ax) (cos(ax) + ax sin(ax))/a2  
sinh x cosh x  
cosh x sinh x  
tanh x ln cosh x  

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