Question

I know that under “with(student)” package, I can solve some integral by parts. I applied this to $int(x*sin(x),x)$ for example and got my answer but, couldn’t use for $int(exp(x)*sin(x),x)$ . I ask if this rule command packaged under with(student) can be applied for a certain kinds of integral or not? Thanks for your help.

Answer 1

with(student):

s1 := Int(exp(x)*sin(x),x) = intparts(Int(exp(x)*sin(x),x),sin(x));

   /                                    /  /                 \
  |                                     | |                  |
  |  exp(x) sin(x) dx = exp(x) sin(x) - | |  cos(x) exp(x) dx|
  |                                     | |                  |
 /                                      \/                   /

s2 := Int(exp(x)*cos(x),x) = expand( intparts(Int(exp(x)*cos(x),x),cos(x)) );

   /                                    /  /                 \
  |                                     | |                  |
  |  cos(x) exp(x) dx = cos(x) exp(x) + | |  exp(x) sin(x) dx|
  |                                     | |                  |
 /                                      \/                   /

s3 := subs(s2,s1);

        /                                                 
       |                                                  
       |  exp(x) sin(x) dx = exp(x) sin(x) - cos(x) exp(x)
       |                                                  
      /                                                   

           /  /                 \
           | |                  |
         - | |  exp(x) sin(x) dx|
           | |                  |
           \/                   /

s4 := lhs(s3) + Int(exp(x)*sin(x),x) = rhs(s3) + Int(exp(x)*sin(x),x);

      /  /                 \                                
      | |                  |                                
    2 | |  exp(x) sin(x) dx| = exp(x) sin(x) - cos(x) exp(x)
      | |                  |                                
      \/                   /                                

lhs(s4)/2 = rhs(s4)/2;

      /                                                     
     |                     1                 1              
     |  exp(x) sin(x) dx = - exp(x) sin(x) - - cos(x) exp(x)
     |                     2                 2              
    /                                                       

…and we’re done.