# How do you evaluate the integral #int x/(root3(x^2-1))#?

The integral equals

Hopefully this helps!

By signing up, you agree to our Terms of Service and Privacy Policy

To evaluate the integral (\int \frac{x}{\sqrt{3(x^2-1)}} , dx), you can use a trigonometric substitution. Let (x = \sec(\theta)), then (dx = \sec(\theta)\tan(\theta) , d\theta). Substituting these into the integral:

[\int \frac{x}{\sqrt{3(x^2-1)}} , dx = \int \frac{\sec(\theta)}{\sqrt{3(\sec^2(\theta)-1)}} \sec(\theta)\tan(\theta) , d\theta]

Simplify the expression inside the square root:

[= \int \frac{\sec(\theta)}{\sqrt{3(\tan^2(\theta))}} \sec(\theta)\tan(\theta) , d\theta]

[= \int \frac{\sec(\theta)}{\sqrt{3\tan^2(\theta)}} \sec(\theta)\tan(\theta) , d\theta]

[= \int \frac{\sec^2(\theta)}{\sqrt{3}\tan(\theta)} , d\theta]

Now, recall that (\sec^2(\theta) - 1 = \tan^2(\theta)). So, (\sqrt{3}\tan(\theta) = \sqrt{\sec^2(\theta) - 1}).

Therefore,

[= \int \frac{\sec^2(\theta)}{\sqrt{\sec^2(\theta) - 1}} , d\theta]

This integral can be simplified using a trigonometric identity. Let (u = \sec(\theta) - \tan(\theta)), then (du = (\sec(\theta)\tan(\theta) + \sec^2(\theta))d\theta).

This leads to:

[= \frac{1}{\sqrt{3}} \int du]

[= \frac{1}{\sqrt{3}}u + C]

Finally, revert back to (x) by substituting (u = \sec(\theta) - \tan(\theta)).

Thus, the solution is:

[= \frac{\sec(\theta) - \tan(\theta)}{\sqrt{3}} + C]

Since we found (x = \sec(\theta)), you can rewrite this in terms of (x):

[= \frac{x}{\sqrt{3(x^2-1)}} + C]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7