a) Find the derivative of coth^ 1(sin2x) . [2] b) Integrate : dxx^3 x^2 2x . [3]

a) Derivative of ${\coth }^{-1}(\sin 2x)$

$$\boxed{\frac{d}{dx}{\coth }^{-1}(\sin 2x)=2\sec 2x}$$

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b) Integration $\int \frac{dx}{x^3-x^2-2x}$

1. Factor the denominator

$$x^3-x^2-2x=x(x^2-x-2)=x(x-2)(x+1)$$

2. Partial‑fraction decomposition

$$\frac{1}{x(x-2)(x+1)}=\frac{A}{x}+\frac{B}{x-2}+\frac{C}{x+1}$$

Multiply by the denominator:

$$1=A(x-2)(x+1)+Bx(x+1)+Cx(x-2)$$

Expanding and equating coefficients gives

$$\{\begin{array}{l}A+B+C=0\\ -A+B-2C=0\\ -2A=1\end{array}⟹A=-\frac{1}{2},B=\frac{1}{6},C=\frac{1}{3}$$

3. Integrate term‑wise

$$\begin{array}{rl}\int \frac{dx}{x^3-x^2-2x}& =\int \left(-\frac{1}{2}\frac{1}{x}+\frac{1}{6}\frac{1}{x-2}+\frac{1}{3}\frac{1}{x+1}\right)dx\\ & =-\frac{1}{2}\ln |x|+\frac{1}{6}\ln |x-2|+\frac{1}{3}\ln |x+1|+C\end{array}$$

$$\boxed{\int \frac{dx}{x^3-x^2-2x}=-\frac{1}{2}\ln |x|+\frac{1}{6}\ln |x-2|+\frac{1}{3}\ln |x+1|+C}$$