sqrt(x)tan^(-1)sqrt(x)

Prove that: i) tan^(-1)(sqrt(x)+sqrt(y))/(1-sqrt(xy))=tan^(-1)sqrt(x)+tan^(-1)sqrt(y) ii) tan^(-1)(x+sqrt(x))/(1-x^(3//2))=tan^(-1)x+tan^(-1)sqrt(x) iii) tan^(-1)(sinx)/(1+cosx)=x/2

[" 0.",int sin^(-1)sqrt((x)/(a+x))dx" is equal to "],[," 1) "(x+a)tan^(-1)sqrt((x)/(a))-sqrt(ax)+C],[" 3) "(x+a)cot^(-1)sqrt((x)/(a))-sqrt(ax)+C," 2) "(x+a)tan^(-1)sqrt((x)/(a))+sqrt(ax)+C]

If intsin^(-1)(sqrt(x)/sqrt(x+1))dx=A(x)tan^(-1)sqrt(x)+B(x)+C then A(x) and B(x) will be

if int tan^(-1)sqrt(x)dx=u tan^(-1)sqrt(x)-sqrt(x)+c then u=

IfI=int(dx)/(x^3sqrt(x^2-1)),t h e nIe q u a l s a. 1/2((sqrt(x^2-1))/(x^3)+tan^(-1)sqrt(x^2-1))+C b. 1/2((sqrt(x^2-1))/(x^2)+xtan^(-1)sqrt(x^2-1))+C c. 1/2((sqrt(x^2-1))/x+tan^(-1)sqrt(x^2-1))+C d. 1/2((sqrt(x^2-1))/(x^2)+tan^(-1)sqrt(x^2-1))+C

IfI=int(dx)/(x^3sqrt(x^2-1)),t h e nIe q u a l s 1/2((sqrt(x^2-1))/(x^3)+tan^(-1)sqrt(x^2-1))+C , 1/2((sqrt(x^2-1))/(x^2)+xtan^(-1)sqrt(x^2-1))+C , 1/2((sqrt(x^2-1))/x+tan^(-1)sqrt(x^2-1))+C , 1/2((sqrt(x^2-1))/(x^2)+tan^(-1)sqrt(x^2-1))+C

IfI=int(dx)/(x^3sqrt(x^2-1)),t h e nIe q u a l s a. 1/2((sqrt(x^2-1))/(x^3)+tan^(-1)sqrt(x^2-1))+C b. 1/2((sqrt(x^2-1))/(x^2)+xtan^(-1)sqrt(x^2-1))+C c. 1/2((sqrt(x^2-1))/x+tan^(-1)sqrt(x^2-1))+C d. 1/2((sqrt(x^2-1))/(x^2)+tan^(-1)sqrt(x^2-1))+C

int(e^(tan^(-1)sqrt(x)))/(sqrt(x)+x sqrt(x))dx

intsqrt(e^x-1)dxi se q u a lto 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)+c sqrt(e^x-1)+tan^(-1)sqrt(e^x-1)+c 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c

intsqrt(e^x-1)dx is equal to (a) 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c (b) sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)+c (c) sqrt(e^x-1)+tan^(-1)sqrt(e^x-1)+c (d) 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c