Antiderivative of log

Antiderivative of log Antiderivative of log To evaluate the integral of logx, we adopt the method of integration by parts.

Integration by parts is particularly helpful to evaluate the integral of a function which is given by the product of two different functions.

Now the question arises, what happens if the function is given alone? In such a case we have to assume the second function equal to 1. Then we apply the method of integration by parts.

The functions like, logx, lnx etc. cannot be integrated alone. Integration by parts uses the application of derivatives to evaluate the integrals of a function. Thus functions, whose derivative is known, can be easily integrated by using integration by parts.

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Now let’s discuss the ant derivative of logx We know the formula of integration by parts for a function f(x) which is the product of two different functions is given by ∫u(x)v(x)dx = u(x) ∫v(x)- ∫[du(x)/dx∫v(x)]dx= first function multiplied by the second function – integral of(derivative of first function into integral of second function.

Now ∫1.log(x), here as we see let’s take log(x) as first and 1 as second function.

Integrating it as per the formula of integration by parts as discussed above we obtain ∫1.log(x),= log(x) ∫1.dx - ∫[(1/x).xdx = logx.x - ∫1.dx =xlogx-x this is the required answer. In the above solution, as we see, first of all we have multiplied the first function log(x) by the integral of second function i.e.1.

Then we have subtracted the integral of the product of derivative of first function and integral of second function as per the formula of integration by parts, and finally we obtain the answer equal to xlogx-x.

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