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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals3 Marks
Question
By using the properties of definite integrals, evaluate the integral $\int\limits_0^{\frac{\pi }{2}} {\frac{{{{\cos }^5}xdx}}{{{{\sin }^5}x + {{\cos }^5}x}}} $
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Answer

Let $I = \int\limits_0^{\frac{\pi }{2}} {\frac{{{{\cos }^5}xdx}}{{{{\sin }^2}x + {{\cos }^5}x}}} dx$…(i)
$ \Rightarrow I = \int\limits_0^{\frac{\pi }{2}} {\frac{{{{\cos }^5}\left( {\frac{\pi }{2} - x} \right)}}{{{{\sin }^5}\left( {\frac{\pi }{2} - x} \right) + {{\cos }^5}\left( {\frac{\pi }{2} - x} \right)}}dx} $
$\left[ {\because \int\limits_0^{{a }{}} {f\left( x \right)dx = \int\limits_0^a {f\left( {a - x} \right)} dx } } \right]$
$\Rightarrow I = \int\limits_0^{\frac{\pi }{2}} {\frac{{{{\sin }^5x}}}{{{{\cos }^5}x + {{\sin }^5}x}}dx} $ …(ii)
Adding equations (i) and (ii),
$2I = \int\limits_0^{\frac{\pi }{2}} {\left( {\frac{{{{\cos }^5}x}}{{{{\cos }^5}x + {{\sin }^5}x}} + \frac{{{{\sin }^5}x}}{{{{\cos }^5}x + {{\sin }^5}x}}} \right)dx} $
$= \int\limits_0^{\frac{\pi }{2}} {\left( {\frac{{{{\cos }^5}x + {{\sin }^5}x}}{{{{\sin }^5}x + {{\cos }^5}x}}} \right)dx} $
$ \Rightarrow 2I = \int\limits_0^{\frac{\pi }{2}} {1dx} $
$\Rightarrow 2I = \frac{\pi }{2}$  
$\Rightarrow I = \frac{\pi }{4}$

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