Q. 337. Integration

Question 337.

Evaluate  ∫ √(5 + x^10) / x^(16) dx.

Answer 337.

Let x^5 = √5 tant

=> 5x^4 dx = √5 sec^2 t dt
=> Integral
= ∫ [5x^4 √(5 + x^10) / 5x^20 dx
= ∫ [(√5 sec^2 t) * √(5 + 5tan^2 t)] dt / 5 * 25tan^4 t
= (1/25) ∫ (sec^3 t / tan^4 t) dt
= (1/25) ∫ (cost / sin^4 t) dt
= (1/25) ∫ sin^-4 t d(sint)

= - 1 / (75sin^3 t) + c ... (1)

 

x^5 = √5 tant

=> tant = x^5/√5

=> sint = x^5 / √(5 + x^10)

=> sin^3 t = x^15 / (5 + x^10)^(3/2)

Plugging in (1),

Integral

= - (5 + x^10)^(3/2) / (75x^15) + c.

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