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integration of log x

The integral of the log function, logx, can be found using integration by parts. The integral of lnx with respect to (x) is:

lnxdx=xlnxx+C

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant.

Proof

To find the integral of the natural logarithm of x, ln(x)dx, we use integration by parts. Integration by parts is based on the product rule for differentiation and is given by:

f(x)g(x)dx=f(x)(g(x)dx){df(x)dxg(x)dx}dx
In our case, we can let f(x)=ln(x) and g(x)=1. Then

  • df(x)dx=1xdx (since the derivative of ln(x) is 1x)
  • g(x)dx=dx=x

Now, substitute these into the integration by parts formula:

ln(x)dx=xln(x)x1xdx

Simplifying the integral on the right:

ln(x)dx=xln(x)1dx

=xln(x)x+C

where C is the constant of integration. So, the integral of ln(x) with respect to x is xln(x)x+C.

Definite Integral of log x

To evaluate a definite integral of the natural logarithm of x, baln(x)dx, where a and b are the limits of integration, we follow a similar process as with the indefinite integral and the we’ll apply the limits at the end.

baln(x)dx=[xln(x)x]ba

=[bln(b)b][aln(a)a]

=bln(b)b(aln(a)a)

=bln(b)baln(a)+a

This is the value of the definite integral of ln(x) from a to b. Note that this formula assumes a and b are in the domain of ln(x), which means a>0 and b>0.

Integration of log 10 x

So far we talked about integration of natural log x i.e logex, now we can check what will be the integration incase of common log x. It is given as

log10xdx=xlog10xxlog10e+C

Integration of log a x

It is given as

logaxdx=xlogaxxlogae+C

Solved Examples

Question 1

Evaluate the definite integral e1ln(x)dx.

Solution

we know ln(x)dx=xln(x)x+C. We apply the limits from 1 to e:

e1ln(x)dx=[xln(x)x]e1=[eln(e)e][1ln(1)1]

Since ln(e)=1 and ln(1)=0, this simplifies to:

=[ee][01]=1

So, e1ln(x)dx=1.

Question 2

Integrate xln(x)

Solution

We will apply the integration by parts here

xln(x)dx=x22ln(x)(x22)(1x)dx=x22ln(x)12xdx=x22ln(x)12x22=x22ln(x)x24

So, the integral of xln(x) with respect to x is x22ln(x)x24+C, where C is the constant of integration.

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