integral of (x+sqrt(1+x^2))^6

解答

\int (x + 1 + x^{2})^{6} d x

\frac{32 x ^{7}}{7} + \frac{48 x ^{5}}{5} + 6 x^{3} + \frac{34}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + x - \frac{2}{7} (x^{2} + 1)^{\frac{7}{2}} - \frac{54}{35} (x^{2} + 1)^{\frac{5}{2}} + 2 (x^{2} + 1)^{\frac{3}{2}} + C

求解步骤

\int (x + 1 + x^{2})^{6} d x

乘开

(x + 1 + x^{2})^{6} : 32 x^{6} + 6 x^{5} x^{2} + 1 + 48 x^{4} + 20 x^{3} (x^{2} + 1)^{\frac{3}{2}} + 18 x^{2} + 6 x (x^{2} + 1)^{\frac{5}{2}} + 1

= \int 32 x^{6} + 6 x^{5} x^{2} + 1 + 48 x^{4} + 20 x^{3} (x^{2} + 1)^{\frac{3}{2}} + 18 x^{2} + 6 x (x^{2} + 1)^{\frac{5}{2}} + 1 d x

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

= \int 32 x^{6} d x + \int 6 x^{5} x^{2} + 1 d x + \int 48 x^{4} d x + \int 20 x^{3} (x^{2} + 1)^{\frac{3}{2}} d x + \int 18 x^{2} d x + \int 6 x (x^{2} + 1)^{\frac{5}{2}} d x + \int 1 d x

\int 32 x^{6} d x = \frac{32 x ^{7}}{7}

\int 6 x^{5} x^{2} + 1 d x = 6 (\frac{1}{7} (1 + x^{2})^{\frac{7}{2}} - \frac{2}{5} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{3} (1 + x^{2})^{\frac{3}{2}})

\int 48 x^{4} d x = \frac{48 x ^{5}}{5}

\int 20 x^{3} (x^{2} + 1)^{\frac{3}{2}} d x = 10 (\frac{2}{5} x^{2} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{35} (x^{2} + 1)^{\frac{7}{2}})

\int 18 x^{2} d x = 6 x^{3}

\int 6 x (x^{2} + 1)^{\frac{5}{2}} d x = 6 (\frac{1}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}})

\int 1 d x = x

= \frac{32 x ^{7}}{7} + 6 (\frac{1}{7} (1 + x^{2})^{\frac{7}{2}} - \frac{2}{5} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{3} (1 + x^{2})^{\frac{3}{2}}) + \frac{48 x ^{5}}{5} + 10 (\frac{2}{5} x^{2} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{35} (x^{2} + 1)^{\frac{7}{2}}) + 6 x^{3} + 6 (\frac{1}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}) + x

化简

\frac{32 x ^{7}}{7} + 6 (\frac{1}{7} (1 + x^{2})^{\frac{7}{2}} - \frac{2}{5} (1 + x^{2})^{\frac{5}{2}} + \frac{1}{3} (1 + x^{2})^{\frac{3}{2}}) + \frac{48 x ^{5}}{5} + 10 (\frac{2}{5} x^{2} (x^{2} + 1)^{\frac{5}{2}} - \frac{4}{35} (x^{2} + 1)^{\frac{7}{2}}) + 6 x^{3} + 6 (\frac{1}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + \frac{1}{7} (x^{2} + 1)^{\frac{5}{2}}) + x : \frac{32 x ^{7}}{7} + \frac{48 x ^{5}}{5} + 6 x^{3} + \frac{34}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + x - \frac{2}{7} (x^{2} + 1)^{\frac{7}{2}} - \frac{54}{35} (x^{2} + 1)^{\frac{5}{2}} + 2 (x^{2} + 1)^{\frac{3}{2}}

= \frac{32 x ^{7}}{7} + \frac{48 x ^{5}}{5} + 6 x^{3} + \frac{34}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + x - \frac{2}{7} (x^{2} + 1)^{\frac{7}{2}} - \frac{54}{35} (x^{2} + 1)^{\frac{5}{2}} + 2 (x^{2} + 1)^{\frac{3}{2}}

解答补常数

= \frac{32 x ^{7}}{7} + \frac{48 x ^{5}}{5} + 6 x^{3} + \frac{34}{7} x^{2} (x^{2} + 1)^{\frac{5}{2}} + x - \frac{2}{7} (x^{2} + 1)^{\frac{7}{2}} - \frac{54}{35} (x^{2} + 1)^{\frac{5}{2}} + 2 (x^{2} + 1)^{\frac{3}{2}} + C

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f (x) = 6 x^{3}

\frac{d}{d z} (\frac{z}{( z ^{2} + R ^{2} ) ^{\frac{3}{2}}})