# Solution to this integral?

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

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

Daniel.

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

dextercioby
Homework Helper
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
Homework Helper
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
Homework Helper
$$C(8)=...?$$
,defining
$$C(x)=:\int_{0}^{x} \cos(t^{2}) dt$$

Daniel.

Galileo
Homework Helper
Approximately 0.68396

dextercioby
Homework Helper

Daniel.

Galileo
That number rounded to 10 decimal places is: $C(8) \approx 0.6839570275$.