Integration by part

Integration by part is implemented via the IPP command. You have to put a defined integral $\int _a^b f(t) \ dt$ at level 2 and a function u(t) at level 1. Let v=f/u', then IPP returns

$$\int _a^b f(t) \ dt =\int _a^b u'v(t) \ dt =[uv(t)]_a^b -\int _a^b u v'(t) \ dt$$

You may call IPP twice or more. In this case, the transformation applies on the first integral of the second member of the equality.
Example:

• 
  
  $$\int _0^x \arcsin(t)^2 \ dt$$
  
  
  'T' IPP $\sqrt{1-T^2}$ IPP EXPA returns the antiderivative of $\arcsin(x)^2$.
  
• Try:
  
  $$\int _0^x \exp(t)\sin(2t) \ dt$$
  
  
  with $\exp(t)$ IPP twice. The resulting equality may be used to compute the antiderivative (RISCH may be called directly on this example, but IPP gives a pedagogical approach)