Example 1: Evaluate \int \csc x dx.
Solution: Note that \csc x = \frac{1}{\sin x} = \frac{\sin^2(x/2)+\cos^2(x/2)}{2\sin(x/2)\cos(x/2)}=\frac{1}{2}\tan(x/2)+\frac{1}{2}\cot{x/2}. Therefore \int\csc xdx = -\ln|\cos(x/2)|+\ln|\sin(x/2)|+C = \ln|\tan(x/2)|+C.
Remark: Note that \tan(x/2)=\frac{1-\cos x}{\sin x}=\csc x - \cot x. \int \csc x dx = \ln|\csc x - \cot x| + C. Moreover, since \ln|\tan(x/2)|=-\ln|\cot(x/2)| and \cot(x/2)=\frac{1+\cos x}{\sin x}=\csc x + \cot x, \int \csc x dx = -\ln|\csc x + \cot x| + C.
Problem 2: Evaluate \int \sqrt{5+4x-x^2} dx.
Hint: Let x-2=3\sin\theta with -\pi/2\leq\theta\leq\pi/2.
Problem 3: Evaluate \int \frac{x^2}{(3+4x-4x^2)^{3/2}} dx.
Hint: Let 2x-1=2\sin\theta with -\pi/2\leq\theta\leq\pi/2.
Problem 4: Evaluate \int x\sqrt{1-x^4}.
Hint: Let u=x^2 and u=\sin \theta.
Problem 5: Evaluate \int_0^{\pi/2}\frac{\cos t}{\sqrt{1+\sin^2 t}}dt.
Hint: Let u = \sin t and u = \tan \theta.
Problem 6: Evaluate \int \frac{x^3+x}{x-1} dx.
Hint: Use long division to get \frac{x^3+x}{x-1} = x^2 + x + 2 + \frac{2}{x-1}.