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Question Number 57235 by maxmathsup by imad last updated on 31/Mar/19

c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{c o s x - s i n x}{\sqrt{{c o s}^{8} x + {s i n}^{8} x}} d x

Commented by maxmathsup by imad last updated on 01/Apr/19

c h a n g e m e n t x = \frac{π}{2} - t g i v e I = - \int_{0}^{\frac{π}{2}} \frac{c o s (\frac{π}{2} - t) - s i n (\frac{π}{2} - t)}{\sqrt{{c o s}^{8} (\frac{π}{2} - t) + {s i n}^{8} (\frac{π}{2} (t)}} (- d t)

= \int_{0}^{\frac{π}{2}} \frac{s i n t - c o s t}{\sqrt{{s i n}^{8} t + {c o s}^{8} t}} d t = - I \Rightarrow 2 I = 0 \Rightarrow I = 0 .

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19

I = \int_{0}^{\frac{π}{2}} \frac{c o s x - s i n x}{\sqrt{{c o s}^{8} x + {s i n}^{8} x}} d x

= \int_{0}^{\frac{π}{2}} \frac{s i n x - c o s x}{\sqrt{{s i n}^{8} x + {c o s}^{8} x}} d x [\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x]

2 I = 0 \to I = 0