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NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If Ι = ∫ dx x^ 3 (x^ 8 + 1)^ 3 / 4 = λ(1 + x^ 8 )(1)/(4) x2 + c...

If

Ι = ∫ \dfrac{dx}{x^{3}\left(x^{8} + 1\right)^{3} / 4} = \dfrac{\lambda\left(1 + x^{8}\right)\dfrac{1}{4}}{x2} + c

(where

c

is the constant of integration), then the value of

\lambda

& is equal to

Options

A $2$
B $\dfrac{1}{2}$
C $- 2$
D $- \dfrac{1}{2}$

Correct answer

D. $- \dfrac{1}{2}$

Step-by-step solution

Ι = ∫ \dfrac{x^{- 9}}{{\left(1 + x^{- 8}\right)}^{\dfrac{3}{4}}}dx

Let,

1 + x^{- 8} = t

- 8x^{- 9}dx = dt

Ι = ∫ \dfrac{- dt}{8t^{3 / 4}}

= - \dfrac{1}{8}∫ t^{- 3 / 4}dt

= - \dfrac{1}{8}\dfrac{t^{1 / 4}}{\dfrac{1}{4}} + c

= - \dfrac{\left(1 + x^{- 8}\right)^{1} / 4}{2} + c

= \dfrac{- \left(1 + x^{8}\right)^{1} / 4}{2x^{2}} + c

⇒ \lambda = - \dfrac{1}{2}

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