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Question
Integrate the function in x cos-1 x.
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Solution
Let `I = int x cos^-1 x dx = int cos^-1 x*x dx`
`= cos^-1 x* int x dx - int [d/dx (cos^-1 x) int x dx] dx`
`= cos^-1 x (x^2/2) - int (-1)/ sqrt (1 - x^2) (x^2/2) dx`
`= x^2/2 cos^-1 x + 1/2 int x^2/ sqrt (1 - x^2) dx`
∴ `I = x^2/2 cos^-1 x+ 1/2 I_1` ....(i)
Where `I_1 = int x^2/ sqrt (1 - x^2) dx`
Put x = cos θ
⇒ dx = -sinθ dθ
∴ `I_1 = int (cos^2 theta (-sin theta))/sqrt (1 - cos^2 theta) d theta`
`= - int cos^2 theta d theta = - 1/2 int (1 + cos 2 theta) d theta`
`= -1/2 (theta + (sin 2 theta)/2) + C`
`= -1/2 (theta + 1/2 xx 2 sin theta cos theta) + C`
`= - 1/2 (theta + cos theta sqrt (1 - cos^2 theta)) + C`
`= - 1/2 (cos^-1 x + x sqrt (1 - x^2)) + C` ....(ii)
From (i) and (ii), we get
`I = (2x^2 - 1) (cos^-1 x)/4 - x/4 sqrt (1 - x^2) + C`
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