Question

integration of logx+logn+log3x​

Answer 1

we have to integrate , logx + logn + log3x

here we have to use integration by part

if two functions v(x) and u(x) are given.

where v(x) is first function in ILATE and u(x) is 2nd function.

then, integration of v(x).u(x) = ∫v(x).u(x).dx = v(x) × ∫u(x).dx - ∫v'(x).(∫u(x).dx).dx

by this concept,

integration of logx = xlogx - x

integration of log3x = xlog3x - x/3

logn is constant term so, integration of logn = logn × x.

hence, integration of (logx + logn + log3x) = (xlogx - x) + logn x + (xlog3x - x/3) + C, where C is constant .

= x(logx + logn + log3x) - x - x/3 + C