Evaluate: (i) `( i i ) (iii)int( i v )1/( v )(( v i )sqrt(( v i i ) (viii)3( i x ))( x )sinx+cosx)( x i ) (xii) dx (xiii)` (xiv) (ii) `( x v ) (xvi)int( x v i i )1/( x v i i i )(( x i x )sin+xsqrt(( x x ) (xxi)3( x x i i ))( x x i i i )cosx)( x x i v ) (xxv) dx (xxvi)` (xxvii)

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)

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)

Evaluate: (i) int(1-sinx)/(x+cosx)\ dx

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)

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

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

Solve ( sin x + "i" cos x )/( 1 +i),i =sqrt(-1) when it is purely imaginary .

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

Evaluate: (i) int(cos^3x)/(sqrt(sinx))\ dx (ii) int(sin^3x)/(sqrt(cosx)\ )\ dx

(i) int ((1+x)^(3))/(sqrt(x))dx " "(ii) int((1+x)^(3))/(x^(4)) dx