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integral of (x^4)/(sqrt(1+x^2))

解答

\int \frac{x ^{4}}{1 + x ^{2}} d x

解答

32 (\frac{3}{256} ln tan (\frac{1}{2} arctan (x)) + 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 )} + \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{1}{2} arctan (x)) - 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 )} - \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{3}} + \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{4}}) + C

求解步骤

\int \frac{x ^{4}}{1 + x ^{2}} d x

使用三角换元法

: \int tan^{4} (u) sec (u) d u

= \int tan^{4} (u) sec (u) d u

使用三角恒等式改写

= \int \frac{sin ^{4} ( u )}{cos ^{5} ( u )} d u

使用换元积分法

: \int \frac{32 v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

= \int \frac{32 v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

提出常数:

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

= 32 \cdot \int \frac{v ^{4}}{( 1 - v ^{2} ) ^{5}} d v

将

\frac{v ^{4}}{( 1 - v ^{2} ) ^{5}}

用部份分式展开:

\frac{3}{256 ( v + 1 )} + \frac{3}{256 ( v + 1 ) ^{2}} - \frac{1}{128 ( v + 1 ) ^{3}} - \frac{3}{64 ( v + 1 ) ^{4}} + \frac{1}{32 ( v + 1 ) ^{5}} - \frac{3}{256 ( v - 1 )} + \frac{3}{256 ( v - 1 ) ^{2}} + \frac{1}{128 ( v - 1 ) ^{3}} - \frac{3}{64 ( v - 1 ) ^{4}} - \frac{1}{32 ( v - 1 ) ^{5}}

= 32 \cdot \int \frac{3}{256 ( v + 1 )} + \frac{3}{256 ( v + 1 ) ^{2}} - \frac{1}{128 ( v + 1 ) ^{3}} - \frac{3}{64 ( v + 1 ) ^{4}} + \frac{1}{32 ( v + 1 ) ^{5}} - \frac{3}{256 ( v - 1 )} + \frac{3}{256 ( v - 1 ) ^{2}} + \frac{1}{128 ( v - 1 ) ^{3}} - \frac{3}{64 ( v - 1 ) ^{4}} - \frac{1}{32 ( v - 1 ) ^{5}} d v

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

= 32 (\int \frac{3}{256 ( v + 1 )} d v + \int \frac{3}{256 ( v + 1 ) ^{2}} d v - \int \frac{1}{128 ( v + 1 ) ^{3}} d v - \int \frac{3}{64 ( v + 1 ) ^{4}} d v + \int \frac{1}{32 ( v + 1 ) ^{5}} d v - \int \frac{3}{256 ( v - 1 )} d v + \int \frac{3}{256 ( v - 1 ) ^{2}} d v + \int \frac{1}{128 ( v - 1 ) ^{3}} d v - \int \frac{3}{64 ( v - 1 ) ^{4}} d v - \int \frac{1}{32 ( v - 1 ) ^{5}} d v)

\int \frac{3}{256 ( v + 1 )} d v = \frac{3}{256} ln ∣ v + 1 ∣

\int \frac{3}{256 ( v + 1 ) ^{2}} d v = - \frac{3}{256 ( v + 1 )}

\int \frac{1}{128 ( v + 1 ) ^{3}} d v = - \frac{1}{256 ( v + 1 ) ^{2}}

\int \frac{3}{64 ( v + 1 ) ^{4}} d v = - \frac{1}{64 ( v + 1 ) ^{3}}

\int \frac{1}{32 ( v + 1 ) ^{5}} d v = - \frac{1}{128 ( v + 1 ) ^{4}}

\int \frac{3}{256 ( v - 1 )} d v = \frac{3}{256} ln ∣ v - 1 ∣

\int \frac{3}{256 ( v - 1 ) ^{2}} d v = - \frac{3}{256 ( v - 1 )}

\int \frac{1}{128 ( v - 1 ) ^{3}} d v = - \frac{1}{256 ( v - 1 ) ^{2}}

\int \frac{3}{64 ( v - 1 ) ^{4}} d v = - \frac{1}{64 ( v - 1 ) ^{3}}

\int \frac{1}{32 ( v - 1 ) ^{5}} d v = - \frac{1}{128 ( v - 1 ) ^{4}}

= 32 (\frac{3}{256} ln ∣ v + 1 ∣ - \frac{3}{256 ( v + 1 )} - (- \frac{1}{256 ( v + 1 ) ^{2}}) - (- \frac{1}{64 ( v + 1 ) ^{3}}) - \frac{1}{128 ( v + 1 ) ^{4}} - \frac{3}{256} ln ∣ v - 1 ∣ - \frac{3}{256 ( v - 1 )} - \frac{1}{256 ( v - 1 ) ^{2}} - (- \frac{1}{64 ( v - 1 ) ^{3}}) - (- \frac{1}{128 ( v - 1 ) ^{4}}))

代回

= 32 \frac{3}{256} ln tan (\frac{arctan ( x )}{2}) + 1 - \frac{3}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 )} - - \frac{1}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{2}} - - \frac{1}{64 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{arctan ( x )}{2}) - 1 - \frac{3}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 )} - \frac{1}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{2}} - - \frac{1}{64 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{3}} - - \frac{1}{128 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{4}}

化简

32 \frac{3}{256} ln tan (\frac{arctan ( x )}{2}) + 1 - \frac{3}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 )} - - \frac{1}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{2}} - - \frac{1}{64 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{a r c t a n ( x )}{2} ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{arctan ( x )}{2}) - 1 - \frac{3}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 )} - \frac{1}{256 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{2}} - - \frac{1}{64 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{3}} - - \frac{1}{128 ( tan ( \frac{a r c t a n ( x )}{2} ) - 1 ) ^{4}} : 32 (\frac{3}{256} ln tan (\frac{1}{2} arctan (x)) + 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 )} + \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{1}{2} arctan (x)) - 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 )} - \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{3}} + \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{4}})

= 32 (\frac{3}{256} ln tan (\frac{1}{2} arctan (x)) + 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 )} + \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{1}{2} arctan (x)) - 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 )} - \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{3}} + \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{4}})

解答补常数

= 32 (\frac{3}{256} ln tan (\frac{1}{2} arctan (x)) + 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 )} + \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{3}} - \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) + 1 ) ^{4}} - \frac{3}{256} ln tan (\frac{1}{2} arctan (x)) - 1 - \frac{3}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 )} - \frac{1}{256 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{2}} + \frac{1}{64 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{3}} + \frac{1}{128 ( tan ( \frac{1}{2} arctan ( x ) ) - 1 ) ^{4}}) + C

作图

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