Reduction formulas for algebraic functions. Hello friends, today I’ll talk about the reduction formulas for exponential and trigonometric functions.

For further posts in reduction formulas in integration:

**Reduction formulas for algebraic functions**

**Solved examples of the reduction formulas for algebraic functions**

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is my first example.

**Example 1**

According to Stroud and Booth (2013)*, “If , obtain a reduction formula for in terms of and hence determine .”

**Solution**

Now here the integral is

And I have to get the reduction formula for . So I’ll start with the * integration by parts*.

**Step 1**

Thus let’s say . So that gives

Therefore will be

and that means

Since , I can say that .

(1)

Also is the integration constant here.

Next, I’ll get the value of .

**Step 2**

Since , I can say that for ,

As I have already got from equation (1), I can say that

(2)

But I don’t know the value of . So for , equation (1) becomes

(3)

Next, I’ll get the value of . Thus for , equation (1) will be

(4)

Now I’ll get the value of . Thus for , equation (1) will be

(5)

Then I’ll get the value of . Since , will be

(6)

Next, I’ll get the values of .

**Step 3**

Now I’ll substitute the value of from the equation (6) to the equation (5) to get

So this gives

Next, I’ll substitute this value of to equation (4) to get

So this gives

Then I’ll substitute this value of to equation (3) to get

Now I’ll simplify it to get

Finally I’ll substitute this value of to equation (2) to get the value of as

where .

Therefore the value of is .

Thus the reduction formula for is . Also, the value of is . Hence I can conclude that these are the answers to the given example.

Now I’ll give you another example.

**Example 2**

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

”

**Solution**

Now here the integral is

And I have to show that

So that means I have to derive the reduction formula for .

As in example 1, here also, I’ll use integration by parts. So that gives

**Step 1**

Thus let’s say that

Therefore will be

So the integration by parts formula gives

Now I’ll simplify it to get

Also, I can rewrite as

So this gives

Therefore becomes

(7)

Also, I already know that Thus will be

Now I’ll substitute these two values of and in equation (7) to get

So this means

Thus will be

Therefore I have already proved the relation. Hence I can conclude that this is the 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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