The integral ∫ 2 x ^ 12 + 5 x ^ 9 ( x ^ 5 + x ^ 3 + 1) ^ 3 dx , is equal to

Correct answer: D. x ^ 10 2 ( x ^ 5 + x ^ 3 + 1) ^ 2 + c Given, I = ∫ 2 x ^ 12 + 5 x ^ 9 ( x ^ 5 + x ^ 3 + 1) ^ 3 dx = ∫ (2 x ^ 12 + 5 x ^ 9 )dx x ^ 15 (1 + 1 x ^ 2 + 1 x ^ 5 ) ^ 3 = ∫ ( 2 x ^ 3 + 5 x ^ 6 )dx (1 + 1 x ^ 2 + 1 x ^ 5 ) ^ 3 Let, 1 + 1 x ^ 2 + 1 x ^ 5 = t ⇒ (- 2 x ^ 3 - 5 x ^ 6 )dx = dt ⇒ ( 2 x ^ 3 + 5 x ^ 6 )dx = - dt I = - ∫ dt t ^ 3 = 1 2 . t ^ 2 I = (1)/(2) 1 (1 + 1 x ^ 2 + 1 x ^ 5 ) ^ 2 + c = (1)/(2) x ^ 10 ( x ^ 5 + x ^ 3 + 1) ^ 2 + c Where c is the constant of integration. = x ^ 10 2 ( x ^ 5 + x ^ 3 + 1) ^ 2 + c

JEE Main 20163 April

Exam: JEE Main
Year: 2016
Date: 3 April
Subject: Mathematics
Chapter: Indefinite Integration

JEE Main 2016 · 3 April · Mathematics · Indefinite Integration

The integral ∫ 2 x ^ 12 + 5 x ^ 9 ( x ^ 5 + x ^ 3 + 1) ^ 3 dx , is...

The integral

∫ \dfrac{2{x}^{12} + 5{x}^{9}}{{\left({x}^{5} + {x}^{3} + 1\right)}^{3}}dx

, is equal to

Options

A $\dfrac{{x}^{5}}{2{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c$
B $\dfrac{- {x}^{10}}{2{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c$
C $\dfrac{- {x}^{5}}{{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c$
D $\dfrac{{x}^{10}}{2{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c$

Correct answer

D. $\dfrac{{x}^{10}}{2{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c$

Step-by-step solution

Given,

I = ∫ \dfrac{2{x}^{12} + 5{x}^{9}}{{\left({x}^{5} + {x}^{3} + 1\right)}^{3}}dx

= ∫ \dfrac{\left(2{x}^{12} + 5{x}^{9}\right)dx}{{x}^{15}{\left(1 + \dfrac{1}{{x}^{2}} + \dfrac{1}{{x}^{5}}\right)}^{3}}

= ∫ \dfrac{\left(\dfrac{2}{{x}^{3}} + \dfrac{5}{{x}^{6}}\right)dx}{{\left(1 + \dfrac{1}{{x}^{2}} + \dfrac{1}{{x}^{5}}\right)}^{3}}

Let,

1 + \dfrac{1}{{x}^{2}} + \dfrac{1}{{x}^{5}} = t

⇒ \left(- \dfrac{2}{{x}^{3}} - \dfrac{5}{{x}^{6}}\right)dx = dt

⇒ \left(\dfrac{2}{{x}^{3}} + \dfrac{5}{{x}^{6}}\right)dx = - dt

I = - ∫ \dfrac{dt}{{t}^{3}} = \dfrac{1}{2 . {t}^{2}}

I = \dfrac{1}{2}\dfrac{1}{{\left(1 + \dfrac{1}{{x}^{2}} + \dfrac{1}{{x}^{5}}\right)}^{2}} + c = \dfrac{1}{2}\dfrac{{x}^{10}}{{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c

Where

c

is the constant of integration.

= \dfrac{{x}^{10}}{2{\left({x}^{5} + {x}^{3} + 1\right)}^{2}} + c

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