Antiderivative of lnx: Formula, Proof

The antiderivative of ln x is equal to x(ln x -1)+C where C is a constant. In this post, we will learn how to find the antiderivative of lnx.

The antiderivative of ln x formula is given by

∫ lnx dx = xln x – x + C,

where C is an integration constant.

Find the Antiderivative of lnx

The antiderivative of lnx is a function of x whose derivative will be lnx. So to get the antiderivative lnx, we need to integrate the function ln x with respect to x. That is, we have to find the integral ∫ lnx dx.

To find the integration of lnx, we will use the integration by parts formula. The formula is used to find the integral of a product function. The formula is

∫uv dx = u∫v dx -∫[$\frac{du}{dx}$∫v dx] dx, where u, v are functions of x.

Putting u=ln x and v=1 in the above formula, we get that

∫ ln x dx

= ∫ (ln x ⋅ 1)dx

= ln x ∫1 dx – $\int \left[\dfrac{d}{dx}(\ln x) \int 1\ dx \right]dx$, applying the above integration by parts formula.

= x lnx – $\int \left[\dfrac{1}{x} \times x \right]dx$ + C

= x lnx – ∫dx + C

= x lnx – x + C

So the antiderivative of lnx is equal to xlnx -x+C where C denotes an integral constant.

Verification:

Now, we show that the derivative of xlnx -x+C is the function lnx. The derivative of xlnx -x+C is equal to

d/dx(xlnx – x +C)

= d/dx(x ln x) – d/dx(x) + d/dx(C)

= x⋅ 1/x + lnx ⋅ 1 – 1+ 0, by the product rule of derivatives. Here we have used the fact that the derivative of a constant is 0 and C is a constant here, so its derivative is zero.

= 1 + ln x -1

= ln x, hence verified.

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