# How do you evaluate the definite integral #int (4x)/sqrt(x^2+1) dx# from # [-sqrt3, sqrt3]#?

We might have spared ourselves the trouble of doing the calculations, noting that:

is an odd function, that is:

So we get:

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

To evaluate the definite integral (\int_{-\sqrt{3}}^{\sqrt{3}} \frac{4x}{\sqrt{x^2+1}} , dx), you can use a trigonometric substitution. Let (x = \tan(\theta)), so (dx = \sec^2(\theta) , d\theta).

Substitute (x) and (dx) in terms of (\theta) into the integral, and adjust the limits accordingly.

[ \int_{-\sqrt{3}}^{\sqrt{3}} \frac{4x}{\sqrt{x^2+1}} , dx = \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{4\tan(\theta)}{\sqrt{\tan^2(\theta) + 1}} \cdot \sec^2(\theta) , d\theta ]

Now, simplify and integrate using trigonometric identities and techniques. After integration, evaluate the antiderivative at the upper and lower limits to find the definite integral value.

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