Get the solution to "Evaluate the integrals. ∫−55w3w50+w20+1"

he298c

Answered question

2021-10-24

Evaluate the integrals.

\int_{- 5}^{5} \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}}

SchulzD

Skilled2021-10-25Added 83 answers

Given integral,

\int_{- 5}^{5} \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}}

Using symmetry propetries
If f is an odd function

(f (- x) = - f (x))

then

\int_{- a}^{a} f (x) d x = 0

f (w) = \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}}

Now

f (- w) = \frac{{(- w)}^{3}}{\sqrt{{(- w)}^{50} + {(- w)}^{20} + 1}}

= \frac{- w^{3}}{\sqrt{w^{50} + w^{20} + 1}}

= - \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}}

= - f (w)

Therefore f is an odd function,
So,

\int_{- 5}^{5} \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}} d w = 0

Final answer:

\int_{- 5}^{5} \frac{w^{3}}{\sqrt{w^{50} + w^{20} + 1}} d w = 0

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?