prove thatintegral a to b ((bx)-1)^(0.5) equalto b[cos^-1(ab)_ ((ab)^(0.5))(1_ (ab))^(0.5)]
shakeel opc , 11 Years ago
Grade 12th Pass
Integral Calculus> prove that integral a to b ((bx)-1)^(0.5)...
1 Answers
Aman Bansal
Last Activity: 11 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 ln(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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