Math  /  Calculus

Question(2xx2+1+1x2+12x1+1(x1)2)\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right)

Studdy Solution
Combine the results:
(2xx2+1+1x2+12x1+1(x1)2)dx=lnx2+1+tan1(x)2lnx11x1+C\int \left(\frac{2x}{x^2+1} + \frac{1}{x^2+1} - \frac{2}{x-1} + \frac{1}{(x-1)^2}\right) \, dx = \ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C
where C=C1+C2+C3+C4 C = C_1 + C_2 + C_3 + C_4 is the constant of integration.
The solution to the integral is:
lnx2+1+tan1(x)2lnx11x1+C\boxed{\ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C}

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