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?

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

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