# Reduction formula problem

1. May 12, 2014

### lionely

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Here is my attempt :

I just can't get the answer the writing near the bottom is me trying to get the left hand side , so you can basically ignore that. But I don't think I made the correct first step... any advice?

Didn't notice the picture was so blurry, the first thing I did was just proceed to integrate In by parts.

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2. May 13, 2014

### BiGyElLoWhAt

Yea your parts wasn't done properly, check it:

$\int_{0}^{\frac{\pi}{2}} e^{2x}sin^nx$
gotta do a parts:
$u = e^{2x}$
$dv = sin^nxdx$
$du=2e^{2x}$
$v = ...$
gotta do another parts and use $sin^2 + cos^2 =1$

$\int_{a}^bsin^nxdx = \int_{a}^b[sin^{n-2}(x)][1-cos^2(x)]dx = \int_{a}^b sin^{n-2}(x)-\int_{a}^{b} sin^{n-2}(x)cos(x)cos(x)$ ...
this just ends up at the reduction formula, and I don't feel like going all the way there, 2 more parts then nest them back together, but the point is: once you get your v
plug it back into the first set of parts and you'll get something really messy that you need to tidy up. I bet that'll simplify to what you have going on in your homework problem.

Last edited: May 13, 2014
3. May 13, 2014

### lionely

I did not think of separating it then making subbing sin^2 x . Thanks.. I'm such an idiot. I just proceeded to integrate by parts without calling sin^n x , sin^(n-2) x* sin^2x

4. May 13, 2014

### lionely

I still don't get you fully, here is my working :

Ignore the first line. where I said In = e^2x sinx^(n-1) * sin^x

I also can't really do parts by the u, dv way or whatever. My teacher said he didn't want me to learn it like that. I just know I should always try to integrate the easier one first and differentiate the other one..

Last edited: May 13, 2014
5. May 13, 2014

### SammyS

Staff Emeritus
What do you mean by
"I also can't really do parts by the u, dv way or whatever." ?

For the second integration it looks like you have:
$u=\sin^{(n-1)}(x)\,\cos(x)\$ and $\ dv=e^{2x}dx \ .$​
Those are good choices.

What is the derivative of
$u=\sin^{(n-1)}(x)\,\cos(x)\ \ ?$​

It's not what you have.

6. May 14, 2014

### lionely

I mean when I carry out integration by parts, I don't call things u or dv or v. I just proceed to integrate. That is how I was taught. The derivative of that would be (n-1)sin^(n-2)xcos^2x -sin^n

7. May 15, 2014

### SammyS

Staff Emeritus
A few parentheses would help here making that derivative:
(n-1)sin(n-2)(x)cos2(x) -sinn(x) .

Notice that cos2(x) = 1 - sin2(x) .

And as you say "integrate the easy part": $\displaystyle \int e^{2x}dx$

Integration by parts for the integral on your second line:
$\displaystyle \int e^{2x}\sin^{n-1}x \cdot \cos x\,dx$​

will give you an integrand with two terms after you simplify.

One of the integrals can be written as In , the other as In-2 .

8. May 15, 2014

### lionely

I got it out finally, I made a mistake with brackets at the 2nd parts... -_-

9. May 15, 2014

### lionely

Thank you guys.