题目 Problem Find ∫4x2+12x dx\begin{align*} \int \frac{4x^2+1}{2\sqrt{x}}\,dx \end{align*}∫2x4x2+1dx giving the answer in its simplest form. (5) 解答 解法一 思路 展开 先把分式拆开，再写成指数形式积分。 答题过程 展开 4x2+12x= 4x22x1/2+12x1/2= 2x3/2+12x−1/2.\begin{align*} \frac{4x^2+1}{2\sqrt{x}} =&\,\frac{4x^2}{2x^{1/2}}+\frac{1}{2x^{1/2}}\\ =&\,2x^{3/2}+\frac12x^{-1/2}. \end{align*}2x4x2+1==2x1/24x2+2x1/212x3/2+21x−1/2. Therefore ∫4x2+12x dx= ∫(2x3/2+12x−1/2) dx= 2⋅x5/25/2+12⋅x1/21/2+c= 45x5/2+x1/2+c= 45x2x+x+c.\begin{align*} \int \frac{4x^2+1}{2\sqrt{x}}\,dx =&\,\int\left(2x^{3/2}+\frac12x^{-1/2}\right)\,dx\\ =&\,2\cdot\frac{x^{5/2}}{5/2} +\frac12\cdot\frac{x^{1/2}}{1/2} +c\\ =&\,\frac45x^{5/2}+x^{1/2}+c\\ =&\,\frac45x^2\sqrt{x}+\sqrt{x}+c. \end{align*}∫2x4x2+1dx====∫(2x3/2+21x−1/2)dx2⋅5/2x5/2+21⋅1/2x1/2+c54x5/2+x1/2+c54x2x+x+c.
