If Φ(x)=∫dxsin12x cos72x, then Φ(π4)−Φ

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Question

If

Φ (x) = \int \frac{d x}{s i n^{\frac{1}{2}} x c o s^{\frac{7}{2}} x}

, then

Φ (\frac{π}{4}) - Φ (0) =

Options:

A . 125

B . 95

C . 65

D . 0

Answer: Option A
:
A

t a n x = t \Rightarrow s e c^{2} x d x = d t ∴ f (x) = \int \frac{d x}{\frac{s i n^{\frac{1}{2}} x}{c o s^{\frac{1}{2}} x} . c o s^{4} x} = \int \frac{(1 + t a n^{2} x) s e c^{2} x}{\sqrt{t a n x}} d x = \int \frac{(1 + t^{2})}{\sqrt{t}} d t = \int (t^{- 1 / 2} + t^{3 / 2}) d t = 2 t^{1 / 2} + \frac{2}{5} t^{5 / 2} = 2 \sqrt{t a n x} + \frac{2}{5} (t a n x)^{5 / 2} ∴ ϕ (\frac{π}{4}) - ϕ (0) = 2 + \frac{2}{5} = \frac{12}{5}

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