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

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), or2 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), orln(tan(x/2)) sin2 x =(1/2)(1-cos2x+sin2x) (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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Thanks

Aman Bansal