If& I = ∫ dx x^ (5)/(2) (1 + x) ^ (7)/(2) 3 = kf(x) + c, where& c is the integration constant and f (1) = (1)/(2(1)/(6)), then the value of& f(2) & is equal to

Correct answer: C. ((2)/(3)) ^ (1)/(6) The given integral is I = ∫ dx [x^ 6 ((1 + x)/(x)) ^ (7)/(2) ] ^ (1)/(3) ⇒ I = ∫ dx x^ 2 (1 + (1)/(x)) ^ (7)/(6) Let, 1 + (1)/(x) = t ⇒ - dx x^ 2 = dt ⇒ I = ∫ - dt t^ (7)/(6) = 6t^ - (1)/(6) + c = 6 ((x)/(x + 1)) ^ (1)/(6) + c ∴ f(x) = ((x)/(x + 1)) ^ (1)/(6) ⇒ f(2) = ((2)/(3)) ^ (1)/(6)

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If& I = ∫ dx x^ (5)/(2) (1 + x) ^ (7)/(2) 3 = kf(x) + c, where& c...

If&

I = ∫ \dfrac{dx}{x^{\dfrac{5}{2}}{\left(1 + x\right)}^{\dfrac{7}{2}}3}= kf\left(x\right) + c,

where&

c

is the integration constant and

f \left(1\right) = \dfrac{1}{2\dfrac{1}{6}},

then the value of&

f\left(2\right)

& is equal to

Options

A6{ $\left(\dfrac{2}{3}\right)}^{\dfrac{1}{6}}$
B6{ $\left(\dfrac{3}{2}\right)}^{\dfrac{1}{6}}$
C{ $\left(\dfrac{2}{3}\right)}^{\dfrac{1}{6}}$
D{ $\left(\dfrac{2}{3}\right)}^{6}$

Correct answer

C. { $\left(\dfrac{2}{3}\right)}^{\dfrac{1}{6}}$

Step-by-step solution

The given integral is

I = ∫ \dfrac{dx}{{\left[x^{6}{\left(\dfrac{1 + x}{x}\right)}^{\dfrac{7}{2}}\right]}^{\dfrac{1}{3}}}

⇒ I = ∫ \dfrac{dx}{x^{2}{\left(1 + \dfrac{1}{x}\right)}^{\dfrac{7}{6}}}

Let,

1 + \dfrac{1}{x} = t

⇒ - \dfrac{dx}{x^{2}} = dt

⇒ I = ∫ - \dfrac{dt}{t^{\dfrac{7}{6}}}

= 6t^{- \dfrac{1}{6}} + c

= 6{\left(\dfrac{x}{x + 1}\right)}^{\dfrac{1}{6}} + c

∴ f\left(x\right) = {\left(\dfrac{x}{x + 1}\right)}^{\dfrac{1}{6}}

⇒ f\left(2\right) = {\left(\dfrac{2}{3}\right)}^{\dfrac{1}{6}}

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