I'm trying to solve∫−∞∞1(4+x2)4+x2dx

Expert answers to "I'm trying to solve∫−∞∞1(4+x2)4+x2dx"

Alessandra Carrillo

Answered question

2022-04-08

I'm trying to solve

\int_{- \infty}^{\infty} \frac{1}{(4 + x^{2}) \sqrt{4 + x^{2}}} d x

Answer & Explanation

oanhtih6

Beginner2022-04-09Added 10 answers

Therefore, after substitution, the integral becomes

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1}{\sqrt{1 + \tan^{2} t}}, d t

To proceed further, you need to use the following cousin of the Pythagorean theorem:

1 + \tan^{2} t = \sec^{2} t

This identity is extremely important and useful in practice -- not the least in manipulating integrals like this. One should be reminded of this identity whenever one comes across an expression like

\sqrt{1 + \tan^{2} t}

\sqrt{1 + x^{2}}

. By the way, the proof of this identity is based on the standard Pythagorean theorem:

1 + \tan^{2} t = 1 + \frac{\sin^{2} t}{\cos^{2} t} = \frac{\cos^{2} t + \sin^{2} t}{\cos^{2} t} = \frac{1}{\cos^{2} t} = \sec^{2} t

From this, it follows that

\frac{1}{\sqrt{1 + \tan^{2} t}} = | \cos t |

Thus the integral becomes

\int_{- \frac{π}{2}}^{\frac{π}{2}} | \cos t |, d t

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Didn't find what you were looking for?