determine the following integrals

Question

∫ (e^2x+1)³ e^2x  dx

Answer 1

∫ (e^2x+1)³ e^2x dx is the problem.

 

Let u = (e^2x+1);

then du = 2e^2xdx  or ½du = e^2xdx,  so you can write the desired integral as

 

½∫u³du and you can take it from here.

 

The trick, often useful, is to see that a substitution can be made making a substitution for a non-trivial expression, recognizing that another factor in an integrand will be essentially the inner (chain rule) derivative.