Can sign of integration by parts equation be changed to negative?

Question

Integration by parts:

$$\int u \:dv+\int v \: du=uv$$

Are there any circumstances under which the sign can be changed so that the following is true?

$$\int u \:dv-\int v \: du=uv$$

Answer 1

$\bullet$ For definite integrals, this is true if $\int_a^b v(t)u'(t) dt=0$

Example in $[-1,1]$ if we put $v(t)=t^2+1$ and $u(t)=\ln(t^2+1)$ then : $$[u(t)v(t)]_{-1}^1=0$$ and :$$\int_{-1}^1udv-\int_{-1}^1 v du = \int_{-1}^1 2t \ln(t^2+1)dt-\int_{-1}^12tdt = 0 $$since $t \mapsto t \ln(t^2+1)$ and $t\mapsto 2t$ are odd one $[-1,1]$

$\bullet$ For indefinite integrals it's true only if $v=0$ or $u$ is constant.

Answer 2

The second statement is true if you have a definite integral and interchange the limits for the second integral, otherwise you cannot just change the signs.