1.
∫√(a^2-x^2) dx, x=asint--> dx=acost dt, cost=√[(1-(x/a)^2]
=∫acost(acost dt)
=∫a^2 cos^2t dt
=(a^2/2)*∫(1+cos(2t)) dt
=(a^2/2)*(t+sin(2t)/2)
=(a^2/2)*{sin-1(x/a)+(x/a)*√[1-(x/a)^2]}+C
2.
∫√(a^2+x^2) dx, x=atant--> dx=asec^2t dt, sect=√[1+(x/a)^2]
=∫asect(asec^2t dt)
=a^2 ∫sec^3t dt
=(a^2/2)(sect*tant+ln|sect+tant|)
=(a^2/2)[(x/a)*√[1+(x/a)^2]+ln|(x/a)+√[1+(x/a)^2]|+C
3.
∫√(x^2-a^2) dx, x=asecx-->dx=asecttant dt
=∫atant (asecttant dt)
=a^2*∫(sec^3-sect)dt
=(a^2/2)*(sect*tant-ln|sect+tant|)
=(a^2/2)*[(x/a)*√[1+(x/a)^2]-ln|(x/a)+√[1+(x/a)^2]|+C