Integration by substitution problem #1?

Question

Here's the first problem I did and don't believe it's right. Could you please help me:

∫sec^2x√tan x dx
u=tan x=sin x/cos x
du/dx=sec^2x
du=sec^2x dx

1/2∫u^1/2 du
1/2√(2/3)u^3/2+c
1/2√(2/3)tan x^3/2+c

If this answer is incorrect, can you please show me the right way of doing this problem. This is a hard section to understand!

Answer 1

Let u=tan(x) then du=sec²(x)dx

∫sec²(x)√tan(x) dx

rearrange the integral

∫√tan(x) sec²(x)dx

This is rewritten in terms of u

∫√u du

The only difference is the factor of ½ you put in front of the integral. Where did you get that from?

∫√u du = 2/3 u^(2/3)

Now substitute u=tan(x) back in.

∫sec²(x)√tan(x) dx = 2/3 tan^(2/3)(x)

Answer 2

If I read your notation correctly, the only mistake I see is that there should not be that factor of 1/2 in front of the integral. With your substitution, du is present in the integral exactly, so there is no need to adjust by a constant multiple.