Integrate (ln x)^n. Hello friends, today I’ll talk about how to integrate (ln x)^n. Have a look!!

Want to know more about the reduction formulas of other functions?? Have a look at the followings:

**Reduction formulas for trigonometric functions**

**Reduction formula for Sin^(n) x | Integrate Sin^(n) x dx****Get the reduction formula for Cos^(n) x | Integrate Cos^(n) x dx****Reduction formula for Tan^(n) x | Integrate Tan^(n) x dx****Integrate Cot^(n) x dx | Reduction formula for Cot^(n) x****Reduction formula for Sec^(n) x | Integrate Sec^(n) x dx**

**Reduction formulas for hyperbolic functions**

**Reduction formulas for algebraic functions**

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### Integrate (ln x)^n

**Solved example of how to integrate (ln x)^n**

Disclaimer: This example is not mine. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is my example.

**Example **

According to Stroud and Booth (2013)*, “If , show that

Hence find

”

**Solution**

Now in this example, I have to integrate the function . Also, I already know that

and I can rewrite it as

So I’ll use the * integration by parts* method.

**Step 1**

And that says

So in this case, I’ll choose as and as .

If , then If , then

Thus becomes

Now I’ll simplify it to get

Since , I can say that

(1)

Hence I have proved the first relation.

Next, I’ll find out the value of .

**Step 2**

If , then is . So I can say that now I’ll get the value of .

Now I’ll substitute in equation (1) to get

(2)

Next, I’ll substitute in equation (1) to get

(3)

Then I’ll substitute in equation (1) to get

(4)

Now for , becomes

So the value of is

where is the integration constant.

Next, I’ll put back in equation (4) to get

Then I’ll substitute in equation (3) to get

Now I’ll simplify it to get the value of as

Next, I’ll substitute this value of in equation (2) to get the value of as

Then I’ll simplify it. And that gives

where is the integration constant.

Thus the value of is .

Hence I can conclude that this is the other solution to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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