JEE Main & Advanced Mathematics Definite Integrals Walli's Formula

Walli's Formula

Category : JEE Main & Advanced

\[\int_{0}^{\pi /2}{{{\sin }^{n}}xdx}=\int_{0}^{\pi /2}{{{\cos }^{n}}xdx}\]

 

 

\[\int_{0}^{\pi /2}{{{\sin }^{n}}xdx}=\int_{0}^{\pi /2}{{{\cos }^{n}}xdx}=\left\{ \begin{matrix} \frac{n-1}{n}.\frac{n-3}{n-2}.\frac{n-5}{n-4}......\frac{2}{3},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{when }n\text{ is odd}  \\ \frac{n-1}{n}.\frac{n-3}{n-2}.\frac{n-5}{n-4}.......\frac{3}{4}.\frac{1}{2}.\frac{\pi }{2},\,\,\,\,\,\text{when }n\text{ is even}  \\ \end{matrix} \right.\]

 

 

\[\int_{0}^{\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}dx}=\frac{(m-1)\,(m-3).....(n-1)\,(n-3)....}{(m+n)\,(m+n-2)\,...(2\text{ or }1)}\],  [If \[m,\,\,n\] are both odd positive integers or one odd positive integer]

 

 

\[\int_{\,0}^{\,\pi /2}{{{\sin }^{m}}x{{\cos }^{n}}xdx}=\frac{(m-1)\,(m-3)............(n-1)\,(n-3)}{(m+n)\,(m+n-2)........(2\text{ or }1)}.\frac{\pi }{2}\], [If \[m,\,\,n\] are both positive integers]


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