Exercise
$\int\ln\frac{\left[1\:+\arccot\left(x\right)\right]}{x^2+1}dx$
Step-by-step Solution
Learn how to solve integrals of rational functions problems step by step online. Solve the integral of logarithmic functions int(ln((1+arccot(x))/(x^2+1)))dx. The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Expand the integral \int\left(\ln\left(1+\mathrm{arccot}\left(x\right)\right)-\ln\left(x^2+1\right)\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\ln\left(1+\mathrm{arccot}\left(x\right)\right)dx results in: \left(\mathrm{arccot}\left(x\right)+1\right)\ln\left(\mathrm{arccot}\left(x\right)+1\right)-\mathrm{arccot}\left(x\right)-1. Gather the results of all integrals.
Solve the integral of logarithmic functions int(ln((1+arccot(x))/(x^2+1)))dx
Final answer to the exercise
$-\mathrm{arccot}\left(x\right)+\left(\mathrm{arccot}\left(x\right)+1\right)\ln\left|\mathrm{arccot}\left(x\right)+1\right|+\left(-x^2-1\right)\ln\left|x^2+1\right|+x^2+C_0$