For n ∈ N , n ≥ 2 , let f ( n ) = ∫ 0 π /4 tan n x d x . Which statement is false?

On ( 0 , π /4 ) , 0 < tan x < 1 , so f ( n + 1 ) < f ( n ) . Also f ( n ) + f ( n − 2 ) = ∫ tan n − 2 x sec 2 x d x = 1/ ( n − 1 ) , implying f ( n ) < 1/ ( n − 1 ) . In particular f ( 3 ) < 1/2 , so (C) is false.

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Mathematics - Definite Integration Question with Solution | TestHub

MathematicsDefinite IntegrationReduction formulaMedium2 minQB

MathematicsMediumsingle choice

For $n \in N, n \geq 2$ , let $f (n) = \int_{0}^{π /4} tan^{n} x d x$ . Which statement is false?

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Answer:

Solution:

On $(0, π /4), 0 < tan x < 1$ , so $f (n + 1) < f (n)$ . Also $f (n) + f (n - 2) = \int tan^{n - 2} x sec^{2} x d x = 1/ (n - 1)$ , implying $f (n) < 1/ (n - 1)$ . In particular $f (3) < 1/2$ , so (C) is false.

Stream:JEESubject:MathematicsTopic:Definite IntegrationSubtopic:Reduction formula

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