# Recitation 5

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 $\frac{x^3+x}{x-1} = x^2 + x + 2 + \frac{2}{x-1}$.