Evaluate: (i) `( i i ) (iii)int( i v )1/( v )(( v i ) (vii) x^(( v i i i )4( i x ))( x )+1)( x i ) (xii) dx (xiii)` (xiv) (ii) `( x v ) (xvi)int( x v i i )(( x v i i i ) (xix) x^(( x x )2( x x i ))( x x i i ))/( x x i i i )(( x x i v ) (xxv) x^(( x x v i )4( x x v i i ))( x x v i i i )+1)( x x i x ) (xxx) (xxxi)` (xxxii) dx (iii) `( i v ) (v)intsqrt(( v i ) (vii)tantheta( v i i i ))( i x )dtheta( x )` (xi)

Convert the following points from polar coordinates to the corresponding Cartesian coordinates. (i) ( i i ) (iii)(( i v ) (v)2, (vi)pi/( v i i )3( v i i i ) (ix) (x))( x i ) (xii) (ii) ( x i i i ) (xiv)(( x v ) (xvi)0, (xvii)pi/( x v i i i )2( x i x ) (xx) (xxi))( x x i i ) (xxiii) (iii) ( x x i v ) (xxv)(( x x v i ) (xxvii)-sqrt(( x x v i i i )2( x x i x ))( x x x ) , (xxxi)pi/( x x x i i )4( x x x i i i ) (xxxiv) (xxxv))( x x x v i ) (xxxvii)

Convert the following polar coordinates to its equivalent Cartesian coordinates. (i) ( i i ) (iii)2,pi)( i v ) (v) (ii) ( v i ) (vii)(( v i i i ) (ix)sqrt(( x )2( x i ))( x i i ) , (xiii)pi/( x i v )6( x v ) (xvi) (xvii))( x v i i i ) (xix) (iii) ( x x ) (xxi)(( x x i i ) (xxiii)-3,-( x x i v )pi/( x x v )6( x x v i ) (xxvii) (xxviii))( x x i x ) (xxx)

If ((1+i)/(1-i))^(x)=1 AA n in N is (i) x = 2n+1 (ii) x =4n (iii) x=2n (iv) x=4n+1

Given points P(2,3),Q(4,-2),a n dR(alpha,0) (i) Find the value of ( i i )alpha( i i i ) (iv) if ( v ) (vi) P R+r Q (vii) (viii) is minimum (ix) Find the value of ( x )alpha( x i ) (xii) if ( x i i i ) (xiv)|( x v ) P R-R Q|( x v i ) (xvii) is maximum

Convert the following Cartesian coordinates to the corresponding polar coordinates using positive r and negative rdot (i) ( i i ) (iii)(( i v ) (v)-1,1( v i ))( v i i ) (viii) (ii) ( i x ) (x)(( x i ) (xii)2,-3( x i i i ))( x i v ) (xv)

(i) int_(0)^(pi//2) x cos x dx (i) int_(1)^(3) x. log x dx

If i= sqrt(-1) , prove that following (x+1+ i) (x + 1- i) (x-1 + i) (x-1-i)= x^(4) + 4

If ((1+i)/(1-i))^(2) - ((1-i)/(1+i))^(2) = x + iy then find (x,y).

Number of four-digit numbers of the form N=a b c d which satisfy following three conditions (i) ( i i ) (iii) 4000lt=N<6000 (iv) (v) (vi) ( v i i ) N (viii) (ix) is a multiple of 5 (x) ( x i ) (xii)3lt=b

If ((1+i)/(1-i))^3-((1-i)/(1+i))^3=x+i y ,\ fin d\ (x , y)