- #1

Machinus

[tex]\int cos(x^2)dx[/tex]

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Machinus
- Start date

- #1

Machinus

[tex]\int cos(x^2)dx[/tex]

- #2

- 13,162

- 725

Daniel.

- #3

- 609

- 0

- #4

- 13,162

- 725

Daniel.

- #5

Galileo

Science Advisor

Homework Helper

- 1,991

- 6

[tex]\cos (x) = \sum_n^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}[/tex]

[tex]\cos (x^2) = \sum_n^{\infty}\frac{(-1)^nx^{4n}}{(2n)!}[/tex]

[tex]\int \cos (x^2)dx = \sum_n^{\infty}\frac{(-1)^nx^{4n+1}}{(4n+1)(2n)!}+C[/tex]

Since the power series for [itex]\cos(x)[/itex] converges for all x, so do the power series for [itex]\cos(x^2)[/itex] and [itex]\int \cos(x^2)dx[/itex].

It may take some computational power if you're interested in values of x that are far from 0.

- #6

- 13,162

- 725

[tex] C(8)=...? [/tex]

,defining

[tex] C(x)=:\int_{0}^{x} \cos(t^{2}) dt [/tex]

Daniel.

- #7

Galileo

Science Advisor

Homework Helper

- 1,991

- 6

Approximately 0.68396

- #8

- 13,162

- 725

Interesting...How many terms did u add??You couldn't have added them all...

Daniel.

Daniel.

- #9

Galileo

Science Advisor

Homework Helper

- 1,991

- 6

That number rounded to 10 decimal places is: [itex]C(8) \approx 0.6839570275[/itex].

That the series converges for all x follows from the ratio test for example.

Share: