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integral of sqrt(16x^4-24x^2+7)x

解答

\int 16 x^{4} - 24 x^{2} + 7 x d x

解答

\frac{1}{8 2 \cdot 2 ^{- \frac{1}{2} + 1}} 42 \cdot 2^{- \frac{1}{2} + 1} x^{2} 16 x^{4} - 24 x^{2} + 7 - 32 \cdot 2^{- \frac{1}{2} + 1} 16 x^{4} - 24 x^{2} + 7 - 4 \cdot 2^{- \frac{1}{2} + 1} x^{2} 8 (x^{2} - \frac{3}{4})^{2} - 1 + 3 16 x^{4} - 24 x^{2} + 7 - 2 \cdot 2^{- \frac{1}{2} + 1} ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1 + C

求解步骤

\int 16 x^{4} - 24 x^{2} + 7 x d x

使用换元积分法

= \int \frac{16 u ^{2} - 24 u + 7}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int 16 u^{2} - 24 u + 7 d u

对

16 u^{2} - 24 u + 7

配方:

16 (u - \frac{3}{4})^{2} - 2

= \frac{1}{2} \cdot \int 16 (u - \frac{3}{4})^{2} - 2 d u

使用换元积分法

= \frac{1}{2} \cdot \int 16 v^{2} - 2 d v

使用分布积分法

= \frac{1}{2} (v 16 v^{2} - 2 - \int \frac{16 v ^{2}}{16 v ^{2} - 2} d v)

\int \frac{16 v ^{2}}{16 v ^{2} - 2} d v = \frac{1}{4} (22 v 8 v^{2} - 1 + ln 22 v + 8 v^{2} - 1)

= \frac{1}{2} (v 16 v^{2} - 2 - \frac{1}{4} (22 v 8 v^{2} - 1 + ln 22 v + 8 v^{2} - 1))

代回

= \frac{1}{2} (x^{2} - \frac{3}{4}) 16 (x^{2} - \frac{3}{4})^{2} - 2 - \frac{1}{4} 22 (x^{2} - \frac{3}{4}) 8 (x^{2} - \frac{3}{4})^{2} - 1 + ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1

化简

\frac{1}{2} (x^{2} - \frac{3}{4}) 16 (x^{2} - \frac{3}{4})^{2} - 2 - \frac{1}{4} 22 (x^{2} - \frac{3}{4}) 8 (x^{2} - \frac{3}{4})^{2} - 1 + ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1 : \frac{1}{8 2 \cdot 2 ^{- \frac{1}{2} + 1}} 42 \cdot 2^{- \frac{1}{2} + 1} x^{2} 16 x^{4} - 24 x^{2} + 7 - 32 \cdot 2^{- \frac{1}{2} + 1} 16 x^{4} - 24 x^{2} + 7 - 4 \cdot 2^{- \frac{1}{2} + 1} x^{2} 8 (x^{2} - \frac{3}{4})^{2} - 1 + 3 16 x^{4} - 24 x^{2} + 7 - 2 \cdot 2^{- \frac{1}{2} + 1} ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1

= \frac{1}{8 2 \cdot 2 ^{- \frac{1}{2} + 1}} 42 \cdot 2^{- \frac{1}{2} + 1} x^{2} 16 x^{4} - 24 x^{2} + 7 - 32 \cdot 2^{- \frac{1}{2} + 1} 16 x^{4} - 24 x^{2} + 7 - 4 \cdot 2^{- \frac{1}{2} + 1} x^{2} 8 (x^{2} - \frac{3}{4})^{2} - 1 + 3 16 x^{4} - 24 x^{2} + 7 - 2 \cdot 2^{- \frac{1}{2} + 1} ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1

解答补常数

= \frac{1}{8 2 \cdot 2 ^{- \frac{1}{2} + 1}} 42 \cdot 2^{- \frac{1}{2} + 1} x^{2} 16 x^{4} - 24 x^{2} + 7 - 32 \cdot 2^{- \frac{1}{2} + 1} 16 x^{4} - 24 x^{2} + 7 - 4 \cdot 2^{- \frac{1}{2} + 1} x^{2} 8 (x^{2} - \frac{3}{4})^{2} - 1 + 3 16 x^{4} - 24 x^{2} + 7 - 2 \cdot 2^{- \frac{1}{2} + 1} ln 22 (x^{2} - \frac{3}{4}) + 8 (x^{2} - \frac{3}{4})^{2} - 1 + C

作图

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