Integration of 1/x | Integral of 1/x

The integration of 1/x is equal to ln|x|+C where ln denotes the natural logarithm, that is, ln|x|=loge|x| and C is the integration constant. In this post, we will learn how to integrate 1/x.

Integration of 1/x Formula

The formula for the integration of 1/x (1 divided by x) is given below.

∫ 1/x dx = ln|x| +C where ln x =logex.

Integration of 1/x

How to Integrate 1/x

Now, we will prove that ∫ 1/x dx = ln|x| +C. Note that 1/x=x-1 cannot be integrated using the power rule integration ∫xn dx = xn+1/(n+1) +C as we have -1+1=0 in the denominator. Thus, to find the integration of 1/x, we will proceed as follows:

Let z=ln x

Differentiating both sides, dz= dx/x. Therefore, we obtain that

∫1/x dx = ∫dx/x = ∫dz

= z+C

= ln x +C

= ln|x|+C as ln is not defined for negative values.

So the integration of 1/x is ln|x|+C where C is the indefinite integral constant.


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Video Solution on Integration of 1/x:

Definite Integral of 1/x

Question: Find the definite integral of 1/x from 1 to 2, that is find ∫12 1/x dx.

As the integral of 1/x is equal to lnx, we obtain that

12 1/x dx = [ln x]12

⇒ ∫12 1/x dx = ln 2 – ln 1

⇒ ∫12 1/x dx = ln 2 – 0 as the value of ln1 is 0.

⇒ ∫12 1/x dx = ln 2.

So the value of the definite integral of 1/x from 1 to 2 is equal to ln2.

FAQs on Integration of 1/x

Q1: What is the integration of 1/x?

Answer: The integration of 1/x is ln|x|+C.

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