Substitution. Consider the integral

Question

HELP PLEASE!

 a) Evaluate the integral by making the substitution

b) Evaluate the integral by making the substitution

c) Use the identity to rewrite the integrand and then evaluate the integral by making the substitution

Answer 1

I'll do the first on for you. Let u = sinh(x) du = cosh(x) dx

Integral becomes int(udu) = u²/2 = 1/2 sinh²(x) + C

The second one is completely symmetric (dcosh(x) = sinh(x)) so you get 1/2 cosh²(x) + C

The third part become int(1/2 sinh(2x)dx) I prefer to see these kind integrals as requiring the derivative of the 2x in front of the dx (i.e. 2dx) so i am integrating

int(1/4 sinh(2x)2dx) and now you get 1/4 cosh(2x) + C (You can also do out the whole substitution - to be honest, I would have done the first two that way as well and just written down the answers. All of these answers are within a constant of each other)