Integrating with Trigonometric Substitution: Solving ∫ x √ 4 + x2 dx

[bengaltiger14]

Homework Statement

Evaluate ∫ x √ 4 + x2 dx by using the trigonometric substitution x = 2tanθ

I am starting on the right track by subbing x=2tanθ into x like this:


=∫ 2tanθ √ 4 + 2tanθ(2)

then, do I just integrate that for the correct answer?

 

[Kurdt]

Just to clear up, are you integrating this:

$$\int \frac{xdx}{\sqrt{4+x^2}}$$

or this:

$$\int x(\sqrt{4+x^2})dx$$

Either way you'll have to put $dx$ in terms of [itex]d\theta[/tex] and then use a trig identity to simplify the whole integral.

 

[bengaltiger14]

$$\int x(\sqrt{4+x^2})dx$$

 

[Kurdt]

Ok, well as I said before, can you write dx in terms of $d\theta$ and substitute for the x's? Thats probably the first step.

 

[George Jones]

If the question requires you to do this integral by trig substitution, then, as Kurdt has said, you have forgotten to change the dx.

If the question allows you to use any method, then tig substitution is not the best method; a different substitution is better. In fact, this integral is simple enough that, after seeing a few more examples of this type, you should be able to write down the answer by inspection.