Question on integral containing exponential and sine function

Question

How to get estimate on following integral:

$$f(x)=\int_c^x\frac{\sin^2(t)}{e^{\sin^2(t)}}dt \,?$$

Here $c$ is positive constant.

I tried doing it so by putting $u= \sin^2(t)$, that way we get:

$$\int\frac{\sqrt{u}}{\sqrt{1-u^2}e^u}dx.$$

But, again I could not get a good estimate.

It would also good if I could get a good asymptotic on it.

Sharp Result of sort $f(x)=g(x)+O(h(x))$ would work for my answer.

Answer 1

We seek to estimate $f(x)=\frac{1}{\exp(\sin^2x)}$ in terms of simple trigonometric functions.

A good estimate of $f(x)$ can be found among the following curves: $$ g(x)=a\cos^4x+b\cos^2x+c $$ where $$g(0)=f(0)\implies a+b+c=1$$and $$ g(\frac{\pi}{2})=f(\frac{\pi}{2})\implies c=e^{-1} $$ hence $$ g(x)=a\cos^4x+(1-e^{-1}-a)\cos^2x+e^{-1} $$ where a good numerical choice for $a$ is $0.3$ and a plot of both functions is as follows

Hence $$ \sin^2x\exp(-\sin^2 x){\approx a\sin^2 x\cos^4 x+b\sin^2 x\cos^2 x+c\sin^2 x \\=0.3\sin^2 x\cos^4 x+0.3321\sin^2 x\cos^2 x+0.3679\sin^2 x. } $$

Answer 2

I shall admit that you want to approximate definite integrals without having recourse to numerical integration.

What you can do is to use $$\frac{\sin^2(x)}{e^{\sin^2(x)}} \sim \frac{x^2-\frac{20 }{1071}x^4 } {1+\frac{1408 }{1071}x^2+\frac{17407 }{32130}x^4 }\qquad \text{for} \qquad 0 \leq x \leq \frac \pi 4$$ $$\frac{\sin^2(x)}{e^{\sin^2(x)}} \sim \frac 1e\,\,\frac{1-\frac{17}{60} \left(x-\frac{\pi }{2}\right)^4 } {1+\frac{13}{60} \left(x-\frac{\pi }{2}\right)^4 }\qquad \text{for} \qquad \frac \pi 4 \leq x \leq \frac \pi 2$$

These are the $[4,4]$ Padé approximants of the integrand respectively built around $x=0$ and $x=\frac \pi 2$. They are easy to integrate.

Now, checking (comparison is done with respect to numerical integration) for the definite integral between $0$ and $a$ $$\left( \begin{array}{ccc} a & \text{approximation} & \text{exact} \\ 0.05\pi & 0.001267 & 0.001267 \\ 0.10\pi & 0.009569 & 0.009569 \\ 0.15\pi & 0.029478 & 0.029478 \\ 0.20\pi & 0.062019 & 0.062030 \\ 0.25\pi & 0.105254 & 0.105341 \\ 0.30\pi & 0.156060 & 0.156148 \\ 0.35\pi & 0.211160 & 0.211248 \\ 0.40\pi & 0.268208 & 0.268297 \\ 0.45\pi & 0.325886 & 0.325974 \\ 0.50\pi & 0.383669 & 0.383757 \end{array} \right)$$