integral of (x^2)/((x^2+8^2)^{3/2)}

解答

\int \frac{x ^{2}}{( x ^{2} + 8 ^{2} ) ^{\frac{3}{2}}} d x

\frac{x ^{3}}{64 ( 64 + x ^{2} ) ^{\frac{1}{2}}} - \frac{1}{64} x (x^{2} + 64)^{\frac{1}{2}} + ln (\frac{1}{8} x + (x^{2} + 64)^{\frac{1}{2}}) + C

求解步骤

\int \frac{x ^{2}}{( x ^{2} + 8 ^{2} ) ^{\frac{3}{2}}} d x

使用分布积分法

= \frac{x ^{3}}{64 ( 64 + x ^{2} ) ^{\frac{1}{2}}} - \int \frac{x ^{2}}{32 ( x ^{2} + 64 ) ^{\frac{1}{2}}} d x

\int \frac{x ^{2}}{32 ( x ^{2} + 64 ) ^{\frac{1}{2}}} d x = \frac{1}{64} x (x^{2} + 64)^{\frac{1}{2}} - ln (\frac{1}{8} x + (x^{2} + 64)^{\frac{1}{2}})

= \frac{x ^{3}}{64 ( 64 + x ^{2} ) ^{\frac{1}{2}}} - (\frac{1}{64} x (x^{2} + 64)^{\frac{1}{2}} - ln (\frac{1}{8} x + (x^{2} + 64)^{\frac{1}{2}}))

化简

= \frac{x ^{3}}{64 ( 64 + x ^{2} ) ^{\frac{1}{2}}} - \frac{1}{64} x (x^{2} + 64)^{\frac{1}{2}} + ln (\frac{1}{8} x + (x^{2} + 64)^{\frac{1}{2}})

解答补常数

= \frac{x ^{3}}{64 ( 64 + x ^{2} ) ^{\frac{1}{2}}} - \frac{1}{64} x (x^{2} + 64)^{\frac{1}{2}} + ln (\frac{1}{8} x + (x^{2} + 64)^{\frac{1}{2}}) + C

\int \frac{1}{15 - 4 x - 4 x ^{2}} d x

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