The value of integral -cosec3 x dx is equal towhere C is integration constant

Let I=∫cosec^3 x dx =∫Icosec xII·cosec^2 x dx Applying by parts, we get I=cosec x( x) ∫ ( cosec x x)( x) dx = cosec x x ∫cosec x(cosec^2 x 1) dx = cosec x ...

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Standard XII

Mathematics

Theorems on Integration

The value of ...

Question

The value of integral $\int {cosec}^{3} x d x$ is equal to
$($ where $C$ is integration constant $)$

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Solution

Let
$I = \int {cosec}^{3} x d x = \int cosec x I \cdot {cosec}^{2} x I I d x$
Applying by parts, we get
$I = cosec x (- cot x) - \int (- cosec x cot x) (- cot x) d x = - cosec x cot x - \int cosec x ({cosec}^{2} x - 1) d x = - cosec x cot x - \int {cosec}^{3} x d x + \int cosec x d x \Rightarrow I = - cosec x cot x - I + ln | cosec x - cot x | + c ∴ I = \frac{- cosec x cot x + ln | cosec x - cot x |}{2} + C$

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Theorems on Integration

Standard XII Mathematics

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The value of integral ∫cosec3 x dx is equal to (where C is integration constant)

The value of integral $\int {cosec}^{3} x d x$ is equal to
$($ where $C$ is integration constant $)$