Q. 2
( − csc 2 x ) + h 3 6 ( 2 csc 2 x cot x ) + … \ln \sin(x+h) = \ln \sin x + h (\cot x) + \frac{h^2}{2} (-\csc^2 x) + \frac{h^3}{6} (2 \csc^2 x \cot x) + \dots ln sin ( x + h ) = ln sin x + h ( cot x ) + 2 h 2 ( − csc 2 x ) + 6 h 3 ( 2 csc 2 x cot x ) + … Simplifying the terms, we get: ln sin ( x + h ) = ln sin x + h cot x − 1 2 h 2 csc 2 x + 1 3 h 3 cot x csc 2 x + … \ln \sin(x+h) = \ln \sin x + h \cot x - \frac{1}{2} h^2 \csc^2 x + \frac{1}{3} h^3 \cot x \csc^2 x + \dots ln sin ( x + h ) = ln sin x + h cot x − 2 1 h 2 csc 2 x + 3 1 h 3 cot x csc 2 x + … 2
- (a)Evaluate lim x → 0 sin x − ln ( e x cos x ) x sin x \lim_{x \to 0} \frac{\sin x-\ln(e^x \cos x)}{x \sin x} lim x → 0 x s i n x s i n x − l n ( e x c o s x ) .
- (b)Find the equation of the asymptotes of 2 x y = .
- (c)Evaluate the integral ∫ x 3 2 x + 3 d x \int x^3 \sqrt{2x+3} dx ∫ x 3 2 x + 3 d x .
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