integral of (3(1+t^{1/2})^2)/(2t^{1/2)}

解答

\int \frac{3 ( 1 + t ^{\frac{1}{2}} ) ^{2}}{2 t ^{\frac{1}{2}}} d t

3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}} + C

求解步骤

\int \frac{3 ( 1 + t ^{\frac{1}{2}} ) ^{2}}{2 t ^{\frac{1}{2}}} d t

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{3}{2} \cdot \int \frac{( 1 + t ^{\frac{1}{2}} ) ^{2}}{t ^{\frac{1}{2}}} d t

乘开

\frac{( 1 + t ^{\frac{1}{2}} ) ^{2}}{t ^{\frac{1}{2}}} : \frac{1}{t ^{\frac{1}{2}}} + 2 + t^{\frac{1}{2}}

= \frac{3}{2} \cdot \int \frac{1}{t ^{\frac{1}{2}}} + 2 + t^{\frac{1}{2}} d t

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{2} (\int \frac{1}{t ^{\frac{1}{2}}} d t + \int 2 d t + \int t^{\frac{1}{2}} d t)

\int \frac{1}{t ^{\frac{1}{2}}} d t = 2 t^{\frac{1}{2}}

\int 2 d t = 2 t

\int t^{\frac{1}{2}} d t = \frac{2}{3} t^{\frac{3}{2}}

= \frac{3}{2} (2 t^{\frac{1}{2}} + 2 t + \frac{2}{3} t^{\frac{3}{2}})

化简

\frac{3}{2} (2 t^{\frac{1}{2}} + 2 t + \frac{2}{3} t^{\frac{3}{2}}) : 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}}

= 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}}

解答补常数

= 3 t^{\frac{1}{2}} + 3 t + t^{\frac{3}{2}} + C

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