\( \int\displaystyle\frac{dx}{\sin^{2}x\cos^{2}x}= \)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \tan x-\cot x+C \) : Where \( C \) is an arbitrary constant
Writing 1 as \( \sin^{2}x+\cos^{2}x \) in the numerator,
\( \displaystyle \int\frac{\sin^{2}x+\cos^{2}x}{\sin^{2}x\cos^{2}x}\,dx=\int\left(\frac{1}{\cos^{2}x}+\frac{1}{\sin^{2}x}\right)dx \)
\( =\int\left(\sec^{2}x+\mathrm{cosec}^{2}x\right)dx \)
\( =\tan x-\cot x+C \)
Tip: \( \int\mathrm{cosec}^{2}x\,dx=-\cot x+C \). Missing this minus sign gives Option (4).