Tiêu đề M1 unit-1_merged.pdf ✅

Unit-I INTEGRAL CALCULUS 2 marks Questions 1) Define Beta and Gamma Functions. 2) Write the relation between Beta and Gamma functions. 3) Write reduction formula for ∫sin m x cos n x dx. 4) Area bounded by the curve y=f (x ), the x-axis and the ordinates x=a , x=b is.... 5) The volume of the solid generated by the revolution about the x-axis of the area bounded the x-axis, of the area bounded by the curve y=f(x), the x-axis and the ordinates x=a, x=b is ....... 4 marks Questions 1) If I n=∫ 0 π 4 tann θ dθ , prove that n ( I n−1+I n+1 )=1. 2) Compute Γ( 3.5) and Γ (-1/2). 3) Prove that β(m+1 , n) m = β (m, n+1) n . 4) Express the integral∫ 0 ∞ a −b x 2 dx in terms of Gamma function. 5) Evaluate ∫ 0 π 6 cos 4 3θ sin3 6θ dθ. 8 marks Questions 1) To show that ∫ 0 π /2 sinn x dx=∫ 0 π / 2 cos n x dx= (n−1) (n−3) (n−5 )..... n (n−2)(n−4 ) ........ ( π 2 ,onlyif n iseven) 2) ∫ 0 π /2 cos m x cos nx dx= m m+n ∫ 0 π /2 cos m−1 x cos(n−1) x dx .Hence deduce that ∫ 0 π /2 cos n x cosnx dx= π 2 n+1 3) Obtain a reduction formula for∫x m (logx) n dx . Hence evaluate ∫ 0 1 x 5 (logx) 3 dx . 4) If n is a positive integer , show that ∫ 0 1 x m (logx) n dx= (−1) n n! (m+1 ) n+1 ,m>−1. 5) Find the area common to the parabola y 2=ax and the circle x 2 + y 2=4 ax . 6) Find whole area of the curve y 2 (a+x )=x 2 (a−x). 7) Find the volume of the reel shaped solid formed by the revolution about the Y-axis, of the part of the parabola y 2=4 ax cut off by the latus rectum. 8) Find the volume of a sphere of radiusa . 9) Prove that β ( m , 1 2)=2 2m−1 β (m,m).
10) Prove that Γ( n+ 1 2)= Γ(2n+1)√π 2 2n Γ (n+1) . 11) Given ∫ 0 ∞ x n−1 1+x dx= π sin nπ , show thatΓ (n) Γ (1−n)= π sin nπ . Hence evaluate ∫ 0 ∞ dy 1+ y 4 . 12) Show that ∫ 0 ∞ e −√x x 7/ 4 dx= 8 3 √π. 13) Show that ∫ 0 π /2 [√tanθ+√secθ] dθ= 1 2 Γ( 1 4 )( Γ ( 3 4 ) +√π /Γ ( 3 4 )) . 14) Show that ∫ 0 π /4 tan7 x dx= 1 12 [5−6 log2]. 15) Find the volume of the solid generated by revolving the ellipse x 2 a 2 + y 2 b 2=1. About (i) the major axis. (ii) the minor the axis.