# Thread: What is the integral of (y^.5)tanydy using integration by parts?

1. ## What is the integral of (y^.5)tanydy using integration by parts?

What is the integral of y1/2tanydy? When I used integration by parts I get a recurring answer that I need to keep repeating integration by parts on.

y1/2(-lncosy)-integral(.5y-.5)(-lncosy)

Then I am stuck. i do not know how to integrate lncosy.

2. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

Are you sure you have the correct function? Also, is this an indefinite integral?

3. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

yes it is the correct function. i'm not sure if it is an indefinite integral or not. antiderivative of square root of y time tangent y dy

4. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

Sorry, but I don't know that this will be integrable. I could be wrong though so hopefully someone with more knowledge comes around.

5. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

Hey gp8283.

Note that -ln(cosy) = ln(1/cosy) = ln(secy). You will need to do integration by parts for this but the derivative of ln(secy) = tan(y).

u = tan(y), du/dy = 1/sec^2(x). y = arctan(u) which means y^(1/2) = SQRT(arctan(u)) tany*cos(y)/cos(y) = sin(y)cos(y)/cos^2(y) = sin(y)cos(y) * sec^2(y) so this integral changes to SQRT(arctan(u))*sin(arctan(u))*cos(arctan(u))*du.

Now you can get formulas for sin(arctan(u)) and cos(arctan(u)) by using Pythagoras's theorem and the trig rations. sin(arctan(u)) = u/SQRT(1 + u^2) with cos(arctan(u)) = SQRT(1 - sin(arctan(u))^2). and y^(1/2) = SQRT(arctan(u)).

I know this is a little out of the way, but the reason I mention this is because if you are solving a recurring integral (like say e^(x)*sin(x)) then you will get an expression somewhere that has the same as your starting integral and then collect both together and simplify.

The integral you have mentioned above has y^(0.5) compared to y^(-0.5) which means that if you keep doing this you won't get the form that you can collect and simpify like the e^(x)*sin(x) example.

Another way that would make things easier is to try the substitution y^(1/2) = tan(u) or even something like sin(u) or cos(u) since there are very specific formulas for getting trig functions in terms of arguments that have arc-sine, arc-cosine, or arc-tangent in them.

6. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

would it be possible to take ((((y^.5tanydy))^2))^.5 square the whole thing and take the square root of that to make y not to the power of 1/2, and just work with ytan^2ydy and then worry about the square root later on antidifferentiation using the chain rule after doing integration by parts on ytan^2dy?

7. ## Re: What is the integral of (y^.5)tanydy using integration by parts?

To clarify, if g(y) = y*tan^2(y), then how are you going to calculate the integral in terms of Integral [g(y)]^(1/2)dy using the chain rule?