Integral problem.Solve the integral f(x)=lnx.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral of ln x can be found using integration by parts.

Int [ u dv ] = u*v - Int[v du]

let u = ln x => du = 1/x

let dv = 1 => v = x

Int [ ln x * 1] = x*ln x - Int[x/x]

=> x*ln x - x + C

=> x(ln x - 1) + C

The integral of ln x  = x(ln x - 1) + C

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll use  integration by parts:

Int f(x)dx = Int ln xdx

We'll apply the formula of integration by parts:

Int udv = uv - Int vdu

u = ln x

We'll differentiate both sides:

du = dx/x

dv = dx => Int dx = Int dv = v

v = x

We'll substitute u,v,du,dv in the formula above:

Int ln xdx = x*ln x - Int xdx/x

We'll simplify and we'll get:

Int ln xdx = x*ln x - Int dx

Int ln xdx = x*ln x - x + C

We'll factorize by x:

Int ln xdx = x(ln x - 1) + C

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