integrate (e^2x) sin(3x+1) dx
by doing integration by parts- can we use e^2x as first function? if then what will be the answer?

Question

integrate (e^2x) sin(3x+1) dx
​by doing integration by parts- can we use e^2x as first
function? if then what will be the answer?

Anuradha Sharma · Accepted Answer

Using ILATE , exponential comes at end so here sin3x+1 will be first function and e2x will be second
So ∫e2xsin3x+1dxI=sin3x+1∫e2xdx-∫ddxsin3x+1∫e2xdxdx=sin3x+1e2x2-32∫e2xcos3x+1dx=sin3x+1e2x2-32cos3x+1∫e2xdx-∫ddxcos3x+1∫e2xdxdx=sin3x+1e2x2-3212e2xcos3x+1+32∫e2xsin3x+1dx=sin3x+1e2x2-34e2xcos3x+1-94II+94I=sin3x+1e2x2-34e2xcos3x+1I = 413sin3x+1e2x2-34e2xcos3x+1

integrate (e^2x) sin(3x+1) dx​by doing integration by parts- can we use e^2x as first function? if then what will be the answer?

integrate (e^2x) sin(3x+1) dx
by doing integration by parts- can we use e^2x as first function? if then what will be the answer?