Evaluate $I = \int x\ln\sqrt{1+x^2}$
Integration by parts yields that
$$I = \frac{1}{4}[(1+x^2)\ln(1+x^2)-x^2]+C_1$$
while integration by substitution $(u=x^2+1)$ shows that
$$I = \frac{1}{4}[(1+x^2)\ln(1+x^2)-x^2-1]+C_2$$
I have checked lots of times but could not find a mistake. Can anyone explain why they are different by $\frac{1}{4}$?
Remember that when evaluating an indefinite integral, we include a constant of integration. Each result obtained through integrating a given function is equivalent up to a constant.
You should have added, e.g., $+C$ to your result. The other result should have included a constant of integration, too, $+C_2$. Then any constants can be added to the constant, to create more simply $C_3 = C_2-\frac 14$.
The results are equivalent, up to the constants of integration. What matters is that in each case, we can differentiate the result to obtain the very same integrand, since the derivative of the constants is zero, in each case.