# Trig substitution integration problem, test in 1hr 30min

√(9-x²) / (x²)

## Homework Equations

Just trig substitution

## The Attempt at a Solution

Ok, for trig sub I did

u=asinΘ
x=3sinΘ
9-x²=9-9sin²=9(1-sin²Θ)
so putting it into the equation

√9cos²Θ=3cosΘ/x^2

where do I go from here, I tried getting help at Math forum, but they were complete tools. Plus I have had a good experience with this site in the past

HallsofIvy
Homework Helper
You do understand, don't you that many of the people who post here, also post at other forums?

Since you say this is an integral, √(9-x²) / (x²) is NOT the problem statement: it has to be something like $\int \sqrt{9- x^2}/x^2 dx$.

Yes, with $x= 3sin(\theta)$ $\sqrt{9- x^2}= 3\sqrt{1- sin^2(\theta)}= 3cos(\theta)$. Now, what is x2 in terms of $\theta$. You need to replace that too. And what is dx? You need to write that in terms of $\theta$.

I still think I am making a mistake somewhere

I did
x²=9sin²Θ

so------- 3cosΘ/9sin²Θ= (1/3)cotΘcscΘ

int(1/3)cotΘcscΘ

=(x/3)ln(sinu)-ln(cscu+cotu)
using the triangle i found
(x/3)ln(x/3)-ln(3/x+(√9-x²/x))

I forgot to take into acount the dx, but when I do i get

(3cosΘ/9sin²Θ)*(3cosΘ)=9cos²Θ/9sin²Θ= cot²

but the integral of cot is ln(sinU) or it can be -ln(cscu)

I am stuck

x = 3siny

dx = 3cosy dy

(9-9siny^2)^1/2. 3cosy dy/9siny^2

(9-9siny^2) = 9cosy^2 because cosy^2 + sin ^2 = 1, multiply both sides by 9.

it becomes

3cosy . 3 cos y dy / 9 siny^2

9cosy^2 dy / 9 siny^2

it becomes cosy^2 dy / sin^2 which is coty^2 dy

but 1 + coty^2 = cscy^2. coty^2 = cscy^2 - 1

now integrate this cscy^2 - 1

you get -cot(y) - y.

I am not so sure about my solution.

The right answer is here, but I'm not sure which steps i am messing up

http://integrals.wolfram.com/index.j...%2F%28x%5E2%29 [Broken]

Integrals can have many forms of answers and this one is one of them but what they share is that they have common output.

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What Racer has done/said is correct; in this case, his solution is right, and integrals can be of different form. I agree after having made an error the first time :S

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