integral of 2(2+x^2)^{3/2}

解答

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

\frac{3}{2} x (x^{2} + 2)^{\frac{1}{2}} + \frac{1}{2} x (x^{2} + 2)^{\frac{3}{2}} + 3 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{1}{2} (x^{2} + 2) + C

求解步骤

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

提出常数:

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

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

使用分布积分法

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

\int 3 x^{2} (2 + x^{2})^{\frac{1}{2}} d x = 12 (- \frac{1}{16} (x (2 + x^{2})^{\frac{1}{2}} + 2 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{2 + x ^{2}}{2}) + \frac{1}{16} x (x^{2} + 2)^{\frac{3}{2}})

= 2 (x (2 + x^{2})^{\frac{3}{2}} - 12 (- \frac{1}{16} (x (2 + x^{2})^{\frac{1}{2}} + 2 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{2 + x ^{2}}{2}) + \frac{1}{16} x (x^{2} + 2)^{\frac{3}{2}}))

化简

2 (x (2 + x^{2})^{\frac{3}{2}} - 12 (- \frac{1}{16} (x (2 + x^{2})^{\frac{1}{2}} + 2 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{2 + x ^{2}}{2}) + \frac{1}{16} x (x^{2} + 2)^{\frac{3}{2}})) : \frac{3}{2} x (x^{2} + 2)^{\frac{1}{2}} + \frac{1}{2} x (x^{2} + 2)^{\frac{3}{2}} + 3 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{1}{2} (x^{2} + 2)

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

解答补常数

= \frac{3}{2} x (x^{2} + 2)^{\frac{1}{2}} + \frac{1}{2} x (x^{2} + 2)^{\frac{3}{2}} + 3 ln \frac{x}{2 ^{\frac{1}{2}}} + \frac{1}{2} (x^{2} + 2) + C

\int x^{2} (x^{3} + 6)^{6} d x

t an g e n t f (x) = 1 + x^{3}, at x = 2

\frac{d y}{d x} = x - 1

f (x) = \frac{6 x ^{2} + 10}{4 x ^{3}}

(e^{t})^{^{'}}