Question

Calculate the definite integral (0,3) of (dx/(x^2+3))

Answer 1

\[\int\limits_{0}^{3} \frac{ dx }{ x ^{2}+3 }\]

Answer 2

Are you expected to know about trig substitution yet or are you supposed to have the inverse trig functions memorized?

Answer 3

At this point I am supposed to use trig substitution. I've tried both \[f'(x) = \frac{ 1 }{ x ^{2}+1 } and f'(x) = \frac{ 1 }{ x }\] but its just not coming out to the right answer.

Answer 4

So what you can do is recognize which version of the pythagorean identity does this look like? \[\LARGE \sin^2 \theta + \cos ^2 \theta =1\] So what we do is start out with this and then we can divide the whole thing by cos^2 theta to get: \[\LARGE \tan^2 \theta + 1 =\sec ^2 \theta \]

So notice that this part has a squared thingy and added to a constant. If we multiply this whole equation by 3 it becomes even more familiar: \[\LARGE 3\tan^2 \theta + 3 =3\sec ^2 \theta \]

So it looks like if we can replace in our integral \[\LARGE x^2=3\tan^2 \theta\] we will be good. So that's what we do, take the square root of both sides of this and then take its derivative to get dx. \[\LARGE x=\sqrt{3}\tan \theta\]\[\LARGE dx=\sqrt{3}\sec^2 \theta d \theta\] So now we can plug these into the integral. Try that out now and I'll help you if you get stuck or if you don't really get why I did some of the steps I did.

Answer 5

I apologize because you wrote all this out, but when I meant substitution I meant where you use the integral chain rule by finding f'(x), g(x), and then du! I'm sorry I should've specified that.. these steps do make sense though!

Answer 6

I am doing that lol.

Answer 7

I've just called it d theta instead of du

Answer 8

\[\LARGE \int\limits_0^3 \frac{dx}{x^2+3}\] So I'm saying now that I've solved for these, I can replace them by substitution:\[\LARGE \int\limits_0^{\pi/4} \frac{\sqrt{3}\sec^2 \theta d \theta}{3\tan^2\theta+3}\]
\[\LARGE \int\limits_0^{\pi/4} \frac{\sqrt{3}\sec^2 \theta d \theta}{3\sec^2\theta} =\LARGE \int\limits_0^{\pi/4} \frac{ d \theta}{\sqrt{3}}\]

Then you plug back in your original substitution when you're done or just evaluate at the new limits of integration.

If you think there's "another way" that's simpler or that I'm somehow doing it "the wrong way" I'm sorry to say that it doesn't exist.