What is the integral of (log(2) x / x)dx?

1 Answer | Add Yours

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the function f(x) = (log(2) x / x)dx.

Now to find the integral first let us convert the log(2) x to log with the base e as that makes things easier to calculate.

Now log (a) b = log (x) b / log (x) a, where x can stand for any base.

So we have log (2) x = ln x / ln 2.

Let y = ln x

=> dy/ dx = 1/x

=> dx / x = dy

Int[(log(2) x / x)dx]

=> Int [ (ln x/ ln 2) / x dx]

=> (1/ ln 2)*Int ( y dy)

=> (1/ ln 2)*y^2 / 2 + C

=> (1 / 2*ln 2)* y^2 + C

substitute y = ln x

=> (1 / 2*ln 2)* (ln x)^2 + C

Therefore the required integral is (1 / 2*ln 2)* (ln x)^2 + C

We’ve answered 318,959 questions. We can answer yours, too.

Ask a question