**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}.