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CBSE Class 12 · 2019 · Indefinite IntegralsShort Answer

Find

\int \frac{s i n ^{3} x + c o s ^{3} x}{s i n ^{2} x c o s ^{2} x} d x

Answer: $sec x - cosec x + C$

Solution

Split the fraction term by term:

\frac{sin ^{3} x}{sin ^{2} x cos ^{2} x} + \frac{cos ^{3} x}{sin ^{2} x cos ^{2} x} = \frac{sin x}{cos ^{2} x} + \frac{cos x}{sin ^{2} x} = sec x tan x + cosec x cot x .

Integrating each standard piece,

\int sec x tan x d x + \int cosec x cot x d x = sec x - cosec x + C .

sec x - cosec x + C

CBSE Class 12 Board · 2019 · Subjective

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