jee-advanced 2003 Q20

jee-advanced · India · screening Integration by Substitution Reduction Formula or Recurrence via Integration by Parts

20. If $I ( m , n ) = \int _ { 0 } { } ^ { 1 } t ^ { m } ( 1 + t ) ^ { n } \mathrm { dt }$, then the expression for $I ( m , n )$ in terms of $I ( m + 1 , n$ 1) is:
(a) $\left. \left. 2 ^ { n } / m + 1 \right) - n / m + 1 \right) * I ( m + 1 , n - 1 )$
(b) $n / m + 1 ) * \mid ( m + 1 , n - 1 )$
(c) $\quad 2 ^ { n } / ( m + 1 ) + n / ( m + 1 ) * I ( m + 1 , n - 1 )$
(b) $\quad m / ( m + 1 ) * I ( m + 1 , n - 1 )$

For every $\alpha \in \left( 0 , \frac { \pi } { 2 } \right)$, the value of $\sqrt { x ^ { 2 } + x } + \frac { \tan ^ { 2 } \alpha } { \sqrt { x ^ { 2 } + x } } , x > 0$ is greater than or equal to

20. If $I ( m , n ) = \int _ { 0 } { } ^ { 1 } t ^ { m } ( 1 + t ) ^ { n } \mathrm { dt }$, then the expression for $I ( m , n )$ in terms of $I ( m + 1 , n$ 1) is:\\
(a) $\left. \left. 2 ^ { n } / m + 1 \right) - n / m + 1 \right) * I ( m + 1 , n - 1 )$\\
(b) $n / m + 1 ) * \mid ( m + 1 , n - 1 )$\\
(c) $\quad 2 ^ { n } / ( m + 1 ) + n / ( m + 1 ) * I ( m + 1 , n - 1 )$\\
(b) $\quad m / ( m + 1 ) * I ( m + 1 , n - 1 )$\\

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