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प्रश्न
Integrate the following w.r.t.x : log (x2 + 1)
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उत्तर
Let I = `int log (x^2 + 1)*dx`
= `int [log (x^2 + 1)]*1dx`
= `[log(x^2 + 1)] int 1dx - int [d/dx{log (x^2 + 1)} int 1dx]*dx`
= `[log (x^2 + 1)]*x - int 1/(x^2 + 1)*dx (x^2 + 1)*xdx`
= `xlog(x^2 + 1) - int (2x^2)/(x^2 + 1)*dx`
= `xlog (x^2 + 1) - int (2x^2 + 2 - 2)/(x^2 + 1)*dx`
= `xlog(x^2 + 1) - int[(2(x^2 + 1))/(x^2 + 1) - 2/(x^2 + 1)]*dx`
= `xlog(x^2 + 1) - int[2 int 1dx - 2 int 1/(x^2 + 1)*dx]`
= x log (x2 + 1) – 2x + 2 tan–1 x + c.
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