theorem:\u00a0<\/b>if v(x)\u00a0is a continuous function with an antiderivative v(x),\u00a0<\/i><\/i>then \u00a0where <\/i>, <\/i>\u00a0are in the domain of v(x).\u00a0<\/i><\/i><\/p>\n

the ftc says that we can pick any old antiderivative v(x)<\/em> for v(x)<\/em>, so we need to compute a string of antiderivatives for the integrands of the terms in the sum. in the previous post we discussed but did not state:<\/p>\n

the power rule:<\/b> the derivative ‘=<\/p>\n

we used this to find that the integral ,\u00a0and since we only need one antiderivative to evaluate definite integrals, we can take \u00a0for use in this case.<\/p>\n

therefore we can evaluate (using the fact that\u00a0,\u00a0,\u00a0\u00a0and the ftc):<\/p>\n