1.∫sec^2 (4x+1)dx; u=4x+1
2.∫ root(sin pi theta) cos(pi theta) d(theta); u= sin(pi theta)
3.∫ e^x dx /(1+e^x); u= 1+e^x
For #1, tan^4(4x+1)/4 +c..is the answer..but I don't get why tan should be ^4??
I took the deriv. of the u and replace dx with the stuff that only contains du..not dx...
And then,,I sometimes have problems solving some questions..
just use the u substitutions you have, find du, and replace them into the integral to get a simpler integral you can do.
for the 2nd one, u=sin(pi theta) and du=(pi)cos(pi theta) d(theta). so replace sin(pi theta) with u so you will have sqrt(u). look at your integral and see that you have cos(pi theta) d(theta) in there. so solve for cos(pi theta) d(theta) and that will equal (1/pi) du and replace cos(pi theta) d(theta) with (1/pi) du and the whole integral will become (1/pi)∫ sqrt(u) du. using the power rule, the integral of u^(1/2) is (2/3)u^(3/2) + C so now just plug u=sin(pi theta) back in to get your final answer.
basically look for a function and its derivative within the integrand and that will be a good candidate for a u-substitution.