# Antiderivative of lnx/x: Formula, Proof | Integral of lnx/x

The antiderivative of lnx/x is equal to 1/2 (lnx)2+C where C is a constant. In this post, let’s learn how to find the antiderivative of ln(x)/x.

The antiderivative formula of lnx/x is given by

∫(lnx)/x dx = 1/2 (lnx)2+C,

where C is an integration constant. This is also the integral formula of lnx divided by x.

## Find Antiderivative of lnx/x

Answer: The antiderivative of ln(x)/x is 1/2 (lnx)2+C.

Proof:

The antiderivative of ln(x)/x is equal to the integral of ln(x)/x. So let us find the integral ∫ln(x)/x dx.

Evaluate ∫ $\dfrac{\ln x}{x}$ dx

Put $\ln x = z$

Differentiating both sides, we get that

$\dfrac{1}{x} = \dfrac{dz}{dx}$

⇒ $\dfrac{dx}{x}$ = dz

Thus the integral of ln(x)/x will be equal to

∫ $\dfrac{\ln x}{x}$ dx

= ∫ $\ln x \dfrac{dx}{x}$ (rewriting the integral)

= ∫ z dz

= z2/2 + C by the power rule of integration.

= $\dfrac{(\ln x)^2}{2}+C$ as z=ln x.

Hence, ∫lnx/x dx = 1/2 (lnx)2+C.

So the antiderivative of ln(x)/x is equal to 1/2 (lnx)2+C where C is a constant on integration.

Read the Antiderivatives of:

## FAQs

### Q1: What is the antiderivative of (lnx)/x?

Answer: The antiderivative of ln(x)/x is equal to 1/2 (lnx)2+C where C is a constant.

### Q2: What is the integral of lnx/x?

Answer: The integral of lnx/x is equal to 1/2 (lnx)2+C.

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