Mathematics

Solve:
$$\int \dfrac{dx}{\sin^2x\cos^2x}$$


SOLUTION
$$\int{\cfrac{dx}{\sin^{2}{x} \cos^{2}{x}}}$$
Multiply and divide the above expression by $$4$$, we haave
$$= \int{\cfrac{4dx}{4 \sin^{2}{x} \cos^{2}{x}}}$$
$$= 4 \int{\cfrac{dx}{{\left( 2 \sin{x} \cos{x} \right)}^{2}}}$$
$$= 4 \int{\cfrac{dx}{\sin^{2}{2x}}} \; \left[ \because \sin{2x} = 2 \sin{x} \cos{x} \right]$$
$$= 4 \int{\csc^{2}{2x} dx}$$
$$= 4 \left( \cfrac{-\cot{2x}}{2} \right) + C \; \left[ \because \int{\csc^{2}{ax} \; dx} = \cfrac{-\cot{ax}}{a} \right]$$
$$= -2 \cot{2x} + C$$
$$\therefore \int{\cfrac{dx}{\sin^{2}{x} \cos^{2}{x}}} = -2 \cot{2x} + C$$
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Subjective Medium Published on 17th 09, 2020
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