Exams Joint Entrance Examination Mathematics Definite Integration Question 236 Question 236 of 246 Medium MCQ
If I ( m , n ) = ∫ 0 1 x m − 1 ( 1 − x ) n − 1 d x , m , n > 0 I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0 I ( m , n ) = ∫ 0 1 x m − 1 ( 1 − x ) n − 1 d x , m , n > 0 , then I ( 9 , 14 ) + I ( 10 , 13 ) I(9,14)+I(10,13) I ( 9 , 14 ) + I ( 10 , 13 ) is
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\[ I ( m , m ) = ∫ 0 1 x m − 1 ( 1 − x ) n − 1 d x Let x = sin 2 θ d x = 2 sin θ cos θ d θ I ( m , n ) = 2 ∫ 0 π / 2 ( sin θ ) 2 m − 1 ( cos θ ) 2 n − 1 d θ I ( 9 , 14 ) + I ( 10 , 13 ) = 2 ∫ 0 π / 2 ( sin θ ) 17 ( cos θ ) 27 d θ + 2 ∫ 0 π / 2 ( sin θ ) 19 ( cos θ ) 25 d θ = 2 ∫ 0 π / 2 ( sin θ ) 17 ( cos θ ) 25 d θ = I ( 9 , 13 ) \begin{aligned}
& \mathrm{I}(\mathrm{~m}, \mathrm{~m})=\int_0^1 \mathrm{x}^{\mathrm{m}-1}(1-\mathrm{x})^{\mathrm{n}-1} \mathrm{dx} \\
& \text { Let } \mathrm{x}=\sin ^2 \theta \quad \mathrm{dx}=2 \sin \theta \cos \theta \mathrm{~d} \theta \\
& \mathrm{I}(\mathrm{~m}, \mathrm{n})=2 \int_0^{\pi / 2}(\sin \theta)^{2 \mathrm{~m}-1}(\cos \theta)^{2 \mathrm{n}-1} \mathrm{~d} \theta \\
& \mathrm{I}(9,14)+\mathrm{I}(10,13)=2 \int_0^{\pi / 2}(\sin \theta)^{17}(\cos \theta)^{27} \mathrm{~d} \theta \\
& +2 \int_0^{\pi / 2}(\sin \theta)^{19}(\cos \theta)^{25} \mathrm{~d} \theta \\
& =2 \int_0^{\pi / 2}(\sin \theta)^{17}(\cos \theta)^{25} \mathrm{~d} \theta \\
& =\mathrm{I}(9,13)
\end{aligned} I ( m , m ) = ∫ 0 1 x m − 1 ( 1 − x ) n − 1 dx Let x = sin 2 θ dx = 2 sin θ cos θ d θ I ( m , n ) = 2 ∫ 0 π /2 ( sin θ ) 2 m − 1 ( cos θ ) 2 n − 1 d θ I ( 9 , 14 ) + I ( 10 , 13 ) = 2 ∫ 0 π /2 ( sin θ ) 17 ( cos θ ) 27 d θ + 2 ∫ 0 π /2 ( sin θ ) 19 ( cos θ ) 25 d θ = 2 ∫ 0 π /2 ( sin θ ) 17 ( cos θ ) 25 d θ = I ( 9 , 13 ) \] Answer the question to unlock