**Question 469.**

**Integrate cosec^5 (5x) dx.**

**Answer 469.**

**Let 5x = u**

**=> 5 dx = du**

**=> Integrand**

**= (1/5) ∫ cosec^5 u du ... ( 1 )**

**∫ cosec^3 u du**

**= ∫ cosecu * cosec^2 u du**

**Using integration by parts**

**= cosecu ∫ cosec^2 u du - ∫ [d/du(cosecu) ∫ cosec^2u du] du**

**= - cosecu cotu - ∫ cosecu cot^2 u du**

**= - cosecu cotu - ∫ cosecu (cosec^2 u - 1) du**

**= - cosecu cotu - ∫ cosec^3 u du + ∫ cosecu du**

**=>**

**2 ∫ cosec^3 u du**

**= - cosecu cotu + ln ltan(u/2)l + 2c**

**=> ∫ cosec^3 u du**

**= - (1/2) cosecu cotu + (1/2) ln ltan(u/2)l + c ... ( 2 )**

**Now, ∫ cosec^5 u du**

**= ∫ cosec^3 u * cosec^2 u du**

**Integrating by parts,**

**= cosec^3 u ∫ cosec^2 u du - ∫ [d/du(cosec^3 u) ∫ cosecu du] du**

**= - cosec^3 u * cotu - ∫ 3 cosec^3 u * cot^2u du**

**= - cosec^3 u * cotu - 3 ∫ cosec^3u (cosec^2u - 1) du**

**= - cosec^3 u * cotu - 3 ∫ cosec^5 u du + 3 ∫ cosec^3 u du**

**=> 4 ∫ cosec^5 u du**

**= - cosec^3 u * cotu + 3 ∫ cosec^3 u du**

**[Plugging the value of ∫ cosec^3 u du from ( 2 ) above]**

**= - cosec^3 u * cotu + 3 [ - (1/2) cosecu cotu + (1/2) ln ltan(u/2)l ] + 4c**

**=> ∫ cosec^5 u du**

**= - (1/4) cosec^3 u * cotu - (3/8) cosecu cotu + (3/8) ln ltan(u/2)l ] + c**

**[Plugging in ( 1 ) above]**

**=> Integrand**

**= - (1/20) cosec^3 (5x) * cot(5x) - (3/40) cosec(5x) cot(5x)**

**+ (3/40) ln ltan(5x/2)l + c'. [c' = c/5]**

**Confirmation that the above answer is correct as verified by Wolfram Alpha:**

**Wolfram Alpha Link**

**Link to YA!**