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cos(t)/sqrt(1+sin^2(t)) dt

I'm supposed to use trig. substitution to find the solution. I started by using the formula a^2+x^2 to get x=atanx. In this case, sin(t)=(1)tan(θ), and so cos(t)dt=sec^2(θ)dθ and so I substituted this into the equation and got:

sec^2(θ)/sqrt(1+tan^2(θ)) -> sec^2(θ)/sqrt(sec^2(θ)) -> sec(θ)

Now, I have the integral of sec(θ)dθ, which equals ln abs(secθ+tanθ). When I take this integral at 0, it turns out to be 0, and so I'm left with the answer being solely the integral at pi/2. The problem is that the secant at pi/2 is infinity and so is the tangent at pi/2, and so the answer ends up being infinity, and this is apparently wrong.

Any help would be greatly appreciated.