The value of the integral Ι = ∫ 2x^ 9 + x^ 10 (x^ 2 + x^ 3 ) ^ 3 dx is equal to (where, C is the constant of integration)

Correct answer: A. x^ 4 2 (1 + x) ^ 2 + C Taking x^ 12 common from denominator, Ι = ∫ 2x^ - 3 + x^ - 2 (x^ - 2 + x^ - 1 ) ^ 3 dx Let, x^ - 2 + x^ - 1 = t ⇒ (- 2x^ - 3 - x^ - 2 )dx = dt ∴ Ι = - ∫ dt t^ 3 = t^ - 2 2 + C = 1 2 (x^ - 2 + x^ - 1 ) ^ 2 + C = x^ 4 2 (1 + x) ^ 2 + C

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The value of the integral Ι = ∫ 2x^ 9 + x^ 10 (x^ 2 + x^ 3 ) ^ 3 dx...

The value of the integral

Ι = ∫ \dfrac{2x^{9} + x^{10}}{{\left(x^{2} + x^{3}\right)}^{3}}

is equal to (where,

C is the constant of integration)

Options

A $\dfrac{x^{4}}{2{\left(1 + x\right)}^{2}} + C$
B $\dfrac{x^{6}}{2{\left(x + 1\right)}^{2}} + C$
C $\dfrac{x^{4}}{{\left(x + 1\right)}^{2}} + C$
D $\dfrac{x^{6}}{2{\left(x + 1\right)}^{3}} + C$

Correct answer

A. $\dfrac{x^{4}}{2{\left(1 + x\right)}^{2}} + C$

Step-by-step solution

Taking

x^{12}

common from denominator,

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

Let,

x^{- 2} + x^{- 1} = t

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

∴ Ι = - ∫ \dfrac{dt}{t^{3}}

= \dfrac{t^{- 2}}{2} + C

= \dfrac{1}{2{\left(x^{- 2} + x^{- 1}\right)}^{2}} + C

= \dfrac{x^{4}}{2{\left(1 + x\right)}^{2}} + C

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