Integral of ln u

Integral calculus is a branch of calculus that deals with integration. If f is a function of real variable x, then the definite integral is given by

int_a^bf(x)dx

Where a and b are intervals of a real line.

Integral of ln u means integration of natural logarithmic (ln) function with respect to the variable ' u '. In this article, we are going to see the list of natural logarithmic integral (ln) rules with few example problems.
List of Natural Logarithmic (ln) Integral Rules:

int 1/u dx = ln |u| + C

int  ln u du = uln u  - u + C

int uln u du = (u^2)/2 ln u  - 1/4 u2 + C

int u2 ln u du = (u^3)/3 ln u - 1/9 u3 + C

int (ln u)2 du = u(ln u)2 - 2uln u + 2u + C

int ((ln u)^n)/u dx = 1/(n + 1) (ln u)n+1 + C

int  (du)/(uln u) = ln (ln u) + C

Learning Natural Logarithmic (ln) Integral Rules with Example Problems:

Example problem 1:

Integrate the function f(u) = 5/u

Solution:

Step 1: Given function

f(u) = 5/u

int f(u) du = int5/u du

Step 2: Integrate the given function f(u) = 5/u with respect to ' u',

int5/u du = 5ln (u)+ C

Example problem 2:

Integrate the function f(u) = log (9u)

Solution:

Step 1: Given function

f(u) = log (9u)

int f(u) du = intlog (9u) du

Step 2: Integrate the given function  with respect to ' u' using integration by parts,

Let x = log (9u)    dv = du

dx = 1/u  du         v = u

int x dv = xv - int v dx

int log (9u) du = ulog (9u) - int u 1/u du

= ulog (9u) - int du

= ulog(9u) - u + C

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Example problem 3:

Integrate the function f(u) = 10log (3u)

Solution:

Step 1: Given function

f(u) = 10log (3u)

int f(u) du = int10log (3u) du

The above function can be written as

int f(u) du = 10intlog (3u) du

Step 2: Integrate the given function  with respect to ' u' using integration by parts,

Let u = log (3u)    dv = du

du = 1/u  du         v = u

int x dv = xv - int v dx

10 int log (3u) du = 10[ulog (3u) - int u 1/u du]

= 10ulog (3u) - 10int du

= 10ulog(3u) - 10u + C