Question

Find the value of \[\int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{{{\sin }^{1000}}xdx}}{{{{\sin }^{1000}}x + {{\cos }^{1000}}x}}} \] is equal to
A) 1000
B) 1
C)$\dfrac{\pi }{2} $
D)$\dfrac{\pi }{4} $

Answer 1

Hint: Here we will solve the problem by using the definite integral property $\int\limits_a^b {f(x)dx = \int\limits_a^b {f(a + b - x)dx} } $

Complete step-by-step answer:
Given value is \[\int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{{{\sin }^{1000}}xdx}}{{{{\sin }^{1000}}x + {{\cos }^{1000}}x}}} \]
Now by using the definite integral property i.e. $\int\limits_a^b {f(x)dx = \int\limits_a^b {f(a + b - x)dx} } $
Let us solve the problem
Now if we consider each term in the numerator and denominator of given value as $f(x)$
Then we write the value as
I = $\int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{{{\sin }^{1000}}(\dfrac{\pi }{2} - x)}}{{{{\sin }^{1000}}(\dfrac{\pi }{2} - x) + {{\cos }^{1000}}(\dfrac{\pi }{2} - x)}}} dx \to 1$

We know that
$\sin (\dfrac{\pi }{2} - x) = \cos x$ $ \Rightarrow {\sin ^{1000}}(\dfrac{\pi }{2} - x) = {\cos ^{1000}}x$
$\cos (\dfrac{\pi }{2} - x) = \sin x \Rightarrow {\cos ^{1000}}(\dfrac{\pi }{2} - x) = \sin x$
From this we can rewrite the value as
$ \Rightarrow \int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{{{\cos }^{1000}}xdx}}{{{{\cos }^{1000}}x + {{\sin }^{1000}}x}}} \to 2$
Now by adding equation 1 and 2 we get the value as
$2I = \int\limits_0^{\dfrac{\pi }{2}} {\dfrac{{{{\cos }^{1000}}x + {{\sin }^{1000}}x}}{{{{\cos }^{1000}}x + {{\sin }^{1000}}x}}} dx$
Here in the above term, numerator and denominator has same value so it get cancels and the value after cancellation is 1
So the term can be written as
$ \Rightarrow 2I = \int\limits_0^{\dfrac{\pi }{2}}{1dx} $
We know that $\int {1dx = x} $ so let us apply the limits for $x$ term
$ \Rightarrow 2I \Rightarrow {[x]_0}^{\dfrac{\pi }{2}}$
$ \Rightarrow 2I = \dfrac{\pi }{2} - 0$
$ \Rightarrow I = \dfrac{\pi }{4}$
Therefore the given value is equals to $\dfrac{\pi }{4}$
Option D is the correct

Note: Make a note that we have to apply definite integral properties for this kind of problem. If needed conversions of values have to be done like the above solution.