data/integrals.jsonl

{"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", "index": 0}
{"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": "$\\boxed{I(x) = \\frac{\\pi}{4}-\\frac{x}{2}\\ln(2)}$", "parameters": "$t;x$", "type": "integrals", "index": 1}
{"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": "$\\boxed{I(x) \\approx \\frac{85}{62}\\sqrt{\\frac{2\\pi}{(2+\\frac{\\pi^2}{4})x}}}$", "parameters": "$t;x$", "type": "integrals", "index": 2}
{"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": "$\\boxed{I(x) = 1 - x - \\frac{x^2}{6} + \\frac{x^4}{120} + \\frac{x^5}{6} - \\frac{x^6}{5040} }$", "parameters": "$x$", "type": "integrals", "index": 3}
{"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": "$\\boxed{\\int_{x}^{\\infty} e^{-a t^b} \\, dt \\sim \\frac{e^{-a x^b}}{a b x^{b-1}}}$", "parameters": "$x;a;b$", "type": "integrals", "index": 4}
{"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": "$\\boxed{I(x) \\sim \\sqrt{\\frac{\\pi}{2x}} e^{-x}}$", "parameters": "$t;x$", "type": "integrals", "index": 5}
{"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": "$\\boxed{-\\frac{1}{x \\ln x}}$", "parameters": "$t;x$", "type": "integrals", "index": 6}
{"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", "index": 7}
{"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": "$\\boxed{I(x) \\sim -\\frac{1}{12x^2} + \\frac{1}{120x^4} - \\frac{1}{252x^6}}$", "parameters": "$x; t$", "type": "integrals", "index": 8}
{"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", "index": 9}
{"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", "index": 10}
{"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": "$\\boxed{y(x)= \\frac{2e^{-9x}}{x}}$", "parameters": "$x$", "type": "integrals", "index": 11}
{"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": "$\\boxed{I(x)\\sim \\frac{\\Gamma(x)}{2x}}$", "parameters": "$x$", "type": "integrals", "index": 12}
{"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": "$\\boxed{I(x) = \\frac{1}{2}x^{-3/2}\\cdot\\Gamma(\\frac{3}{4})}$", "parameters": "$x$", "type": "integrals", "index": 13}