Integral of $\frac{1}{x}$

Q: What is the integral of 1 / x?

The integral of (1)/(x) is ln ≤ft(≤ft|x|) + C.

The integral of $\frac{1}{x}$ is $\ln\left(\left|x\right|\right) + C$ .

\int \frac{1}{x}\,dx = \ln\left(\left|x\right|\right) + C

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Graph of f(x) = 1 / x with the area from 1 to 2 shaded. The shaded area equals the definite integral, about 0.69314718.

How to work it out

1
We want an antiderivative of f(x) = 1 / x.
$\int \frac{1}{x}\,dx$
2
Method: standard integral.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int \frac{1}{x}\,dx = \ln\left(\left|x\right|\right) + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\ln\left(\left|x\right|\right)\right] = \frac{\operatorname{sign}\left(x\right)}{\left|x\right|}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 1 and 2:
$\int_{1}^{2} \frac{1}{x}\,dx = \Big[\ln\left(\left|x\right|\right)\Big]_{1}^{2} \approx 0.69314718$

LaTeX source of the formulas on this page

\[ \int \frac{1}{x}\,dx = \ln\left(\left|x\right|\right) + C \]
\[ \int \frac{1}{x}\,dx \]
\[ \frac{d}{dx}\left[\ln\left(\left|x\right|\right)\right] = \frac{\operatorname{sign}\left(x\right)}{\left|x\right|} \]
\[ \int_{1}^{2} \frac{1}{x}\,dx = \Big[\ln\left(\left|x\right|\right)\Big]_{1}^{2} \approx 0.69314718 \]

Definite integrals of the same function

Interval	Value
from 1 to 2	0.69314718
from 1 to 1.5	0.40546511

Questions

What is the integral of 1 / x?

The integral of $\frac{1}{x}$ is $\ln\left(\left|x\right|\right) + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of 1 / x from 1 to 2?

About 0.69314718.

Integral of 1x\frac{1}{x}x1​

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $\frac{1}{x}$