Question:

$ \int_{0}^{\pi /2}{\frac{{{\sin }^{100}}x}{{{\sin }^{100}}x+{{\cos }^{100}}x}\,\,dx} $ is equal to

Updated On: Jun 23, 2024
  • $ \frac{\pi }{2} $
  • $ \frac{\pi }{12} $
  • $ \frac{\pi }{4} $
  • $ \frac{\pi }{8} $
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The Correct Option is C

Solution and Explanation

$ \int_{0}^{\pi /2}{\frac{{{\sin }^{100}}\,x}{{{\sin }^{100}}\,x+\,{{\cos }^{100}}x}}\,\,dx $
$ =\int_{0}^{\pi /2}{\frac{{{\sin }^{100}}}{{{\sin }^{100}}x+{{\sin }^{100}}\left( \frac{\pi }{2}-x \right)}}\,\,dx $
$ =\frac{\frac{\pi }{2}-0}{2}=\frac{\pi }{4} $
$ \left[ \because \,\,\int_{a}^{b}{\frac{f(x)\,dx}{f(x)+f(a+b-x)}=\frac{b-a}{2}} \right] $
Alternate $ I=\int_{0}^{\pi /2}{\frac{{{\sin }^{100}}x}{{{\sin }^{100}}x+{{\cos }^{100}}x}\,\,\,dx} $ .. (i)
$ I=\int_{0}^{\pi /2}{\frac{{{\sin }^{100}}\,\left( \frac{\pi }{2}-x \right)}{{{\sin }^{100}}\left( \frac{\pi }{2}-x \right)-{{\cos }^{100}}\left( \frac{\pi }{2}-x \right)}}\,\,\,dx $
$ \left[ \begin{align} & \because \,\,by\,\,define\,\,\text{integral property}\text{.} \\ & \int_{a}^{0}{f(x)\,dx=\int_{0}^{a}{f(a-x)\,dx}} \\ \end{align} \right] $
$ I=\int_{0}^{\pi /2}{\frac{{{\cos }^{100}}x}{{{\cos }^{100}}x+{{\sin }^{100}}x}}\,dx $ .. (ii)
On adding Eqs. (i) and (ii), we get
$ 2I=\int_{0}^{\pi /2}{1\,\,dx\,=\frac{\pi }{2}-0\,\,\Rightarrow \,I=\frac{\pi }{4}} $
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Concepts Used:

Integrals of Some Particular Functions

There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.

Integrals of Some Particular Functions:

  • ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
  • ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
  • ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
  • ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
  • ∫1/√(a2 – x2) dx = sin-1(x/a) + C
  • ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C

These are tabulated below along with the meaning of each part.