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

解答

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

\frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3) + C

求解步骤

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

化简

x^{2} : xx

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

使用分布积分法

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

\int 3 x^{2} (3 + x^{2})^{\frac{1}{2}} d x = 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}})

= x (3 + x^{2})^{\frac{3}{2}} - 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}})

化简

x (3 + x^{2})^{\frac{3}{2}} - 27 (- \frac{1}{24} (x (3 + x^{2})^{\frac{1}{2}} + 3 ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{3 + x ^{2}}{3}) + \frac{1}{36} x (x^{2} + 3)^{\frac{3}{2}}) : \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3)

= \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3)

解答补常数

= \frac{9}{8} x (x^{2} + 3)^{\frac{1}{2}} + \frac{1}{4} x (x^{2} + 3)^{\frac{3}{2}} + \frac{27}{8} ln \frac{x}{3 ^{\frac{1}{2}}} + \frac{1}{3} (x^{2} + 3) + C

x \to 5 lim (\frac{2}{5 - x})

\frac{d}{d x} ((1 + \frac{8}{x})^{x})

\frac{d}{d x} (x + 1 - x)

\frac{d y}{d t} + e^{t} y = - 4 e^{t}

x \to 0 + lim (\frac{- 4 x ^{2} - 6 x}{x})