Definite Integrals

Question

1.  ¹_{0 ∫√x √1+x √x dx}
2.  ^π/3_{π/6 ∫ (1-cos 0) csc}² _0 d 0
3.  ^e₁ ∫lnx/x dx

Answer 1

i have them all. here is the last

ln(x)/x

u = ln(x)

du = 1/x

dv = 1/x

v = ln(x)

uv - int(vdu) -> ln(x)^2 - int(ln(x)/x)

int(ln(x)/x) = ln(x)^2 - int(ln(x)/x)

2*int(ln(x)/x) = ln(x)^2

= ln(x)^2 / 2. then evaluate