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  "body": "\n## Proof\n\nWe use that the exponential function is an antiderivative of itself.\n\n1. Let $F = \\exp$, the [[exponential function|0RMVS7RXZX]], and write $e^x = \\exp(x)$.\n2. By [[the derivative of exp|0EX9GFJ0B3]], $F$ is differentiable on $\\R$ and $F' = F$.\n3. [[Differentiable functions are continuous|07ZMX4ANJS]], so $F$ is continuous on $\\R$.\n4. Assume $a \u003c b$. By step 3, $F$ is [[Riemann integrable|06SESV7GQY]] on $[a, b]$. By steps 2 and 3, the [[second fundamental theorem of calculus|0XYAFCN0PT]] applies to $F$ with $F' = F$. It gives $\\int_a^b e^x \\, dx = e^b - e^a$.\n5. Assume $a = b$. By the convention for the [[definite integral|021WEM680W]], $\\int_a^a e^x \\, dx = 0 = e^a - e^a$.\n6. Assume $a \u003e b$. By the same convention and step 4 on $[b, a]$, $\\int_a^b e^x \\, dx = -(e^a - e^b) = e^b - e^a$.\n7. So $\\int_a^b e^x \\, dx = e^b - e^a$ for all real $a$ and $b$.\n\n## Idea\n\nThe exponential function is its own derivative, so it is its own antiderivative. The fundamental theorem of calculus then evaluates the integral as a difference of two values of $e^x$.\n",
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