Evaluate $lim_{x \to 0} \frac{t a n x - s i n x}{x ^{3}}$ .

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Differentiate $y = arctan (\frac{2 x}{1 - x ^{2}})$ with respect to $x$ for $∣ x ∣ < 1$ .
Evaluate the definite integral $I = \int_{0}^{π /2} \frac{s i n x}{s i n x + c o s x} d x$ .
Find $\int \frac{1}{x x ^{2} - 1} d x$ for $x > 1$ .
Prove that $sin (3 θ) = 3 sin θ - 4 sin^{3} θ$ . What is the value of $sin (1 8^{\circ})$ derived from…
Find the maximum value of $f (θ) = 3 sin θ + 4 cos θ$ and the smallest positive $θ$ at which it occurs.