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

解答

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

x (2 + x)^{6} - \frac{1}{7} (5 x^{\frac{7}{2}} + 252 x^{\frac{5}{2}} + 560 x^{\frac{3}{2}} + 56 x^{3} + 560 x^{2} - 448 x) + C

求解步骤

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

使用分布积分法

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

\int \frac{6 x ( x + 2 ) ^{5} - ( 2 + x ) ^{6}}{2 x} d x = \frac{1}{7} (5 x^{\frac{7}{2}} + 252 x^{\frac{5}{2}} + 560 x^{\frac{3}{2}} + 56 x^{3} + 560 x^{2} - 448 x)

= x (2 + x)^{6} - \frac{1}{7} (5 x^{\frac{7}{2}} + 252 x^{\frac{5}{2}} + 560 x^{\frac{3}{2}} + 56 x^{3} + 560 x^{2} - 448 x)

解答补常数

= x (2 + x)^{6} - \frac{1}{7} (5 x^{\frac{7}{2}} + 252 x^{\frac{5}{2}} + 560 x^{\frac{3}{2}} + 56 x^{3} + 560 x^{2} - 448 x) + C

\int \frac{1}{2 + 4 x ^{4}} (16 x^{3}) d x

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