利用微積分基本定理以求定積分的關(guān)鍵是求出被積函數(shù)的原函數(shù),即尋找滿足的函數(shù) .如何求出一個(gè)被積函數(shù)的原函數(shù)呢?我們知道求一個(gè)函數(shù)的原函數(shù)與求一個(gè)函數(shù)的導(dǎo)數(shù)是互逆運(yùn)算,所以要求被積函數(shù)的原函數(shù),首先要明確它們之間的關(guān)系:原函數(shù)的導(dǎo)數(shù)就是被積函數(shù),并且導(dǎo)函數(shù)是唯一確定的,而被積函數(shù)的原函數(shù)是不唯一的.即若,則被積函數(shù)的原函數(shù)為(為常數(shù)).
類型一 被積函數(shù)為基本初等函數(shù)的導(dǎo)數(shù)
求這種類型被積函數(shù)的原函數(shù),關(guān)鍵是要記準(zhǔn)上述基本初等函數(shù)的導(dǎo)數(shù)公式,找到對(duì)應(yīng)的被積函數(shù).由基本初等函數(shù)的導(dǎo)數(shù)公式可知:若是被積函數(shù),為原函數(shù),則有: