Integration of log x

What is Integration of log x ?

The integration of log x is given as:

The integration of log x is calculated by the technique (or method) of Integration by Parts (also known as Partial Integration or Product Rule of Integration).

To learn more about it visit our blog on Integration by Parts.

Derivation

By using the formula of integration by parts or by the method of product rule of integration, we will obtain the log x integration.

As we know that the formula of integration by parts is:

∫f(x)g(x).dx = f(x)∫g(x).dx − ∫(f′(x)∫g(x).dx).dx + C

And, we can write logx as (logx.1)

So, ? ∫logx.dx = ∫logx.1.dx

Now,

By using the ILATE Rule,

Let, f(x) = logx and g(x) = 1.

? f’(x) = 1/x.

And,
? ∫g(x).dx = ∫1.dx
? ∫g(x).dx = x.
Then,
By using integration by parts,

=> ∫logx.dx = logx.x – ∫(1/x).x.dx
? ∫logx.dx = x.logx – ∫1.dx
=> ∫logx.dx = x.logx – x + C
? ∫logx.dx = x.(logx – 1) + C
Therefore,
Integration of Log x is ∫logx.dx = x.(logx – 1) + C ; where C = Integration constant.

Examples using Integration of log x

Let’s take a look at some more illustrative examples which consist of logx along with some modifications and variations.

Example-1: Solve: ∫log(x/10).dx

Solution:
By using the formula of integration by parts, we will obtain the integration of log(x/10).
As we know that the formula of integration by parts is:
∫f(x)g(x).dx = f(x)∫g(x).dx − ∫(f′(x)∫g(x).dx).dx + C
And, we can write log(x/10) as:
? log(x/10) = [log(x/10).1]
So,
? ∫log(x/10).dx = ∫log(x/10).1.dx
Now,
By using the ILATE Rule,
Let, f(x) = log(x/10) and g(x) = 1.
? f’(x) = [1/(x/10)].(1/10)
? f’(x) = 1/x.
And,
? ∫g(x).dx = ∫1.dx
? ∫g(x).dx = x.
Then,
By using integration by parts,
? ∫log(x/10).dx = log(x/10).x – ∫(1/x).x.dx
=>∫log(x/10).dx = x.log(x/10) – ∫1.dx
? ∫log(x/10).dx = x.log(x/10) – x + C
=>∫log(x/10).dx = x.(log(x/10) – 1) + C
Therefore,
Answer: ∫log(x/10).dx = x.(log(x/10) – 1) + C ; where C = Integration constant.

Example-2: Find out the integral of x logx.

Solution:
By using the formula of integration by parts, we will calculate the integration of x logx.
As we know that the formula of integration by parts is:
∫f(x)g(x).dx = f(x)∫g(x).dx − ∫(f′(x)∫g(x).dx).dx + C
Now,
By using the ILATE Rule,
Let, f(x) = logx and g(x) = x.
? f’(x) = 1/x.
And,
? ∫g(x).dx = ∫x.dx
? ∫g(x).dx = x²/2.
Then,
By using integration by parts,
? ∫x.logx.dx = logx.(x²/2) – ∫(1/x).(x²/2).dx
=>∫x.logx.dx = (x²/2).logx – ∫(x/2).dx
? ∫x.logx.dx = (x²/2).logx – (1/2).(x²/2) + C
=>∫x.logx.dx = (x²/2).logx – (x²/4) + C
Hence, ∫x.logx.dx = (x²/2).logx – (x²/4) + C.
Answer: ∫x.logx.dx = (x²/2).logx – (x²/4) + C ; where C = Integration constant

Example-3: Calculate the integral of (logx)².

Solution:
By using the formula of integration by parts, we will find out the integration of (logx)2.
As we know that the formula of integration by parts is:
∫f(x)g(x).dx = f(x)∫g(x).dx − ∫(f′(x)∫g(x).dx).dx + C
And, we can write (logx)² as 1.(logx)²
So,
∫(logx)².dx = ∫1.(logx)²dx

Now,
By using the integration by parts and ILATE Rule,

Thus,
Answer: ∫(logx)².dx = x.(logx)² – 2x.logx + 2x + C ; where C = Integration constant.

Example-4: Solve the integration of log(x+1).

Solution:
By using the formula of integration by parts, we will obtain the integration of log(x+3).
As we know that the formula of integration by parts is:
∫f(x)g(x).dx = f(x)∫g(x).dx − ∫(f′(x)∫g(x).dx).dx + C
And, we can write log(x+3) as [log(x+3).1]
So,
∫log(x+3).dx = ∫[log(x+3).1].dx
Now,
By using the ILATE Rule,
Let, f(x) = log(x+3) and g(x) = 1.
? f’(x) = 1/(x+3).
And,
? ∫g(x).dx = ∫1.dx
? ∫g(x).dx = x.
Then,
By using integration by parts,

Therefore,
Answer: ∫log(x+3).dx = (x + 3).log(x+3) – x + C ; where C = Integration constant.

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