Convergence and Solution of Integral with Cosine Function

[Machinus]

$$\int cos(x^2)dx$$

 

[dextercioby]

Not an elementary function.Search the same site (A&S online,see the other thread) for FRESNEL INTEGRALS.

Daniel.

 

[vincentchan]

forget it, this one has no elementary solution.. you can espand the cosine by Taylor series... and integrate the individual term...

 

[dextercioby]

However,the series obtained has a very small convergence radius.It would virtually do you no good.If u have definite integrals involving C(x),then learn they are tabulated.

Daniel.

 

[Galileo]

The power series expansion is the basic surefire way to get a numerical approximation.

$$\cos (x) = \sum_n^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$$

$$\cos (x^2) = \sum_n^{\infty}\frac{(-1)^nx^{4n}}{(2n)!}$$

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

Since the power series for $\cos(x)$ converges for all x, so do the power series for $\cos(x^2)$ and $\int \cos(x^2)dx$.
It may take some computational power if you're interested in values of x that are far from 0.

 

[dextercioby]

Please,Galileo,compute using your formula
$$C(8)=...?$$
,defining
$$C(x)=:\int_{0}^{x} \cos(t^{2}) dt$$

Daniel.

 

[Galileo]

Approximately 0.68396

 

[dextercioby]

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


Daniel.

 

[Galileo]

I just used maple to sum the thing from n=0 to n=100.
That number rounded to 10 decimal places is: $C(8) \approx 0.6839570275$.

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