QJEEM19Ma88c7: JEE Main 2019 Mathematics - Class 12 JEEM19Ma88c7 Solution: The integral ∫ s e c 2 / 3 x c o s e c 4 / 3 x d x is equal to…

Question

Verified Solution • 2025–26 Session moderate

The integral

\int s e c^{2 / 3} x c o s e c^{4 / 3} x d x

is equal to (Hence C is a constant of integration)

Accepted Answer

\int \sec^{\frac{2}{3}} x \cos e c^{\frac{4}{3}} x d x

=

\int \frac{\sec^{\frac{2}{3}} x}{\cos e c^{\frac{2}{3}} x} \cos e c^{2} x d x

=

\int \frac{1}{\cot^{\frac{2}{3}} x} \cos e c^{2} x d x

Let cot x = t³

\Rightarrow

- cosec²x dx = 3t²dt

=

- 3 \int \frac{t^{2} d t}{t^{2}}

= -3t + C

= -3

\cot^{\frac{1}{3}} x

+ C

= -3

\tan^{- \frac{1}{3}} x

+ C

Step-by-Step Verified Solution