prompt: Consider the following integral:$\int_0^{5} ( \frac{e^{-x}}{1 + x^2}) e^{-\epsilon (\frac{\sin^2(x)}{1 + x^4})} dx$In the limit$\epsilon \rightarrow \infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{\sqrt{\frac{2 \pi}{2 \epsilon}}}$

parameters: $x; \epsilon$

type: integrals


prompt: Consider the following integral:$I(x) = \int_0^1[\frac{e^{-xt}}{1+t^2}]dt$In the limit$x \rightarrow 0$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) = \frac{\pi}{4}-\frac{x}{2}\ln(2)}$

type: integrals


prompt: Consider the following integral:$I(x) = \int_1^\infty g(t) e^{-xf(t)}dt; g(x)=\frac{85}{-t+t^6}; f(t) = (\ln(t-1))^2 + \cos(\frac{\pi}{2} t) + 1$In the limit$x \rightarrow \infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) \approx \frac{85}{62}\sqrt{\frac{2\pi}{(2+\frac{\pi^2}{4})x}}}$

type: integrals


prompt: Consider the following integral:$I(x)=\int_x^{1}cos(xt)dt$In the limit$x \to 0+$, find approximate behavior of the integral up to and including the order x^6. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) = 1 - x - \frac{x^2}{6} + \frac{x^4}{120} + \frac{x^5}{6} - \frac{x^6}{5040} }$

type: integrals


prompt: Consider the following integral:$I(x) = \int_{x}^{\infty} e^{-at^b} dt$In the limit$x \to +\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide an expression for the approximate behavior of the integral in a $\boxed{}$ latex environment.

solution: $\boxed{\int_{x}^{\infty} e^{-a t^b} \, dt \sim \frac{e^{-a x^b}}{a b x^{b-1}}}$

type: integrals


prompt: Consider the following integral:$ I(x) = \int_{x}^{\infty} K_0(t) \, dt $In the limit$x \to +\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}}$

type: integrals


prompt: Consider the following integral:$\int_{0}^{1/e} \frac{e^{-xt}}{\ln t} \, dt$In the limit$x \to +\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{-\frac{1}{x \ln x}}$

type: integrals


prompt: Consider the following integral:$I(\epsilon) = \int_0^{10} \frac{1}{(\epsilon + 4x^3 + 2x^9)^{3/2}} dx$In the limit$\epsilon \rightarrow \infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(\epsilon) = \frac{1}{\epsilon^{3/2}} \cdot 10}$

parameters: $\epsilon$

type: integrals


prompt: Consider the following integral:$I(x) = -\int_{0}^{\infty} \left[ \frac{1}{e^t - 1} - \frac{1}{t} + \frac{1}{2} \right] e^{-xt} \, dt$In the limit$x \to +\infty$, find the asymptotic expansion of the integral up to and including the first three leading orders in z. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) \sim -\frac{1}{12x^2} + \frac{1}{120x^4} - \frac{1}{252x^6}}$

type: integrals


prompt: Consider the following integral:$I(\epsilon) = \int_0^{10} \frac{dx}{(\epsilon + 9x^5 + x^{11})^\frac{13}{7}}$In the limit$\epsilon \to \infty$, find approximate behavior of the integral up to and including the first leading order in \epsilon. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(\epsilon) = 10\cdot\epsilon^{-13/7}}$

parameters: $\epsilon$

type: integrals


prompt: Consider the following integral:$I(\epsilon) = \int_0^{10} \frac{dx}{(\epsilon + 9x^5 + x^{11})^\frac{13}{7}}$In the limit$\epsilon \to 10^6$, find approximate behavior of the integral up to and including the first leading order in \epsilon. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(\epsilon) = \frac{\sqrt[11]{-1 + 2^{\frac{7}{13}}}}{\epsilon^{\frac{136}{77}}}}$

parameters: $\epsilon$

type: integrals


prompt: Consider the following integral:$I(x) = \int_0^3 (\cos(t^2) + 5 + 2t^3) e^{-x(2e^t + 7 + \sin(t))} dt$In the limit$x\to\infty$, find approximate behavior of the integral up to and including the first leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{y(x)= \frac{2e^{-9x}}{x}}$

type: integrals


prompt: Consider the following integral:$I(x)=\int_0^\infty \frac{t^{x-1}e^{-t}}{t+x}dt$In the limit$x\to\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x)\sim \frac{\Gamma(x)}{2x}}$

type: integrals


prompt: Consider the following integral:$I(x) = \int_0^{π/4}\sqrt{sin (t)}e^{-x^2t^2}dt$ In the limit$x \rightarrow \infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.

solution: $\boxed{I(x) = \frac{1}{2}x^{-3/2}\cdot\Gamma(\frac{3}{4})}$

type: integrals