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.