While trying to solve the indefinite integral of (x+1)/((x^2 + 1)^2), I end up having to solve the integral of arctan(x) + x/(x^2 + 1)
I rewrite this as the sum ot two integrals, and since the integral of arctan(x) = x*arctan(x) - integral of x/(x^2 + 1), the integral of arctan(x) + x/(x^2 + 1) = x*arctan(x)
The calculator ends up with the answer x*arctan(x) + (x^2)/(x^2 +1) + 1/(x^2 + 1) by integration by parts, which in geogebra looks like the same answer differing by a constant. However, this difference causes me to end up with an entirely different answer for the main integral to the one in the calculator, which is the same as the solution on the answer sheet. All my other steps are the same as the ones in the calculator. Why does it make such a differnce? How can I know which way will lead me to the correct answer?
I rewrite this as the sum ot two integrals, and since the integral of arctan(x) = x*arctan(x) - integral of x/(x^2 + 1), the integral of arctan(x) + x/(x^2 + 1) = x*arctan(x)
The calculator ends up with the answer x*arctan(x) + (x^2)/(x^2 +1) + 1/(x^2 + 1) by integration by parts, which in geogebra looks like the same answer differing by a constant. However, this difference causes me to end up with an entirely different answer for the main integral to the one in the calculator, which is the same as the solution on the answer sheet. All my other steps are the same as the ones in the calculator. Why does it make such a differnce? How can I know which way will lead me to the correct answer?
