We can integrate logx using the method integration by parts.
The formula for integration by parts :
∫udv = uv - ∫vdu
Integration of logx is
∫logxdx
In ∫logxdx, consider logx as u and dx as dv.
That is,
u = logx
dv = dx
Find du and v.
u = logx du/dx = 1/x du = (1/x)dx |
dv = dx ∫dv = ∫dx v = x |
∫logxdx = uv - ∫vdu
Substitute u = lox, v = x and du = (1/x)dx.
∫logxdx = (logx)(x) - ∫(x)(1/x)dx
∫logxdx = xlogx - ∫dx
∫logxdx = xlogx - x + c
∫logxdx = x(logx - 1) + c
So, integration of lox is equal to x(logx - 1) + c.
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