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Question
Integrate the function in tan-1 x.
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Solution
Let `I = int tan^-1 x dx`
`= int tan^-1 x. 1 dx`
Put `u = tan^-1 x, v = 1`
`int uv dx = u int v dx - int ((du)/dx int v dx) dx`
`I= int tan^-1 x. 1`
`(tan^-1 x) int 1 dx - (d/dx (tan^-1 x) int dx) dx`
`= x tan^-1 x - int 1/(1 + x^2) . x dx`
`= x tan^-1 x - 1/2 int (2x)/(1 + x^2) dx`
Put 1 + x2 = t, and dx = dt
`= x tan^-1 x - 1/2 int dt/t`
`= x tan^-1 x - 1/2 log t + C`
`= x tan^1 - 1/2 log (1 + x^2) + C`
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