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Question: If f(x) = \(\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x}\) then f \(\left( \frac{\pi}{4} \right)\) – ...

If f(x) = ∫dxsin⁡1/2xcos⁡7/2x\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x} then f (π4)\left( \frac{\pi}{4} \right) – f(0) is equal to

A

2.2

B

2.3

C

2.4

D

2.5

Answer

2.4

Explanation

Solution

f(x) = ∫dxsin⁡1/2xcos⁡1/2x.cos⁡4x\int_{}^{}\frac{dx}{\frac{\sin^{1/2}x}{\cos^{1/2}x}.\cos^{4}x}

f(x) = ∫sec⁡4xtan⁡x\int_{}^{}\frac{\sec^{4}x}{\sqrt{\tan x}} dx =∫(1+tan⁡2x)sec⁡2xdxtan⁡x\int_{}^{}\frac{(1 + \tan^{2}x)\sec^{2}xdx}{\sqrt{\tan x}}

f(x) = ∫1+t2t\int_{}^{}\frac{1 + t^{2}}{\sqrt{t}}dt = ∫(t−1/2+t3/2)dt\int_{}^{}{(t^{- 1/2} + t^{3/2})}dt

f(x) = 2t1/2 + 2/5 t5/2 = 2tan⁡x2\sqrt{\tan x} + 2/5 (tan x)5/2

f (π4)\left( \frac { \pi } { 4 } \right) – f(0) = 2 + 2/5 = 2.4