integral of x/((sqrt(7x-10-x^2))^3)

Lösung

\int \frac{x}{( 7 x - 10 - x ^{2} ) ^{3}} d x

4 (2 (\frac{( 2 x - 7 ) ^{2}}{72 - x ^{2} + 7 x - 10} + \frac{1}{18} - x^{2} + 7 x - 10) + \frac{7 ( 2 x - 7 )}{36 - x ^{2} + 7 x - 10}) + C

Schritte zur Lösung

\int \frac{x}{( 7 x - 10 - x ^{2} ) ^{3}} d x

Vervollständige das Quadrat

7 x - 10 - x^{2} : - (x - \frac{7}{2})^{2} + \frac{9}{4}

= \int \frac{x}{( - ( x - \frac{7}{2} ) ^{2} + \frac{9}{4} ) ^{3}} d x

Wende U-Substitution an

= \int \frac{4 ( 2 u + 7 )}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u

Entferne die Konstante:

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

= 4 \cdot \int \frac{2 u + 7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u

Multipliziere aus

\frac{2 u + 7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} : \frac{2 u}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} + \frac{7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}}

= 4 \cdot \int \frac{2 u}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} + \frac{7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{2 u}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u + \int \frac{7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u)

\int \frac{2 u}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u = 2 (\frac{u ^{2}}{9 - 4 u ^{2} + 9} + \frac{1}{36} - 4 u^{2} + 9)

\int \frac{7}{( - 4 u ^{2} + 9 ) ^{\frac{3}{2}}} d u = \frac{7 u}{9 - 4 u ^{2} + 9}

= 4 (2 (\frac{u ^{2}}{9 - 4 u ^{2} + 9} + \frac{1}{36} - 4 u^{2} + 9) + \frac{7 u}{9 - 4 u ^{2} + 9})

Setze in

u = x - \frac{7}{2}

ein

= 4 2 \frac{( x - \frac{7}{2} ) ^{2}}{9 - 4 ( x - \frac{7}{2} ) ^{2} + 9} + \frac{1}{36} - 4 (x - \frac{7}{2})^{2} + 9 + \frac{7 ( x - \frac{7}{2} )}{9 - 4 ( x - \frac{7}{2} ) ^{2} + 9}

Vereinfache

4 2 \frac{( x - \frac{7}{2} ) ^{2}}{9 - 4 ( x - \frac{7}{2} ) ^{2} + 9} + \frac{1}{36} - 4 (x - \frac{7}{2})^{2} + 9 + \frac{7 ( x - \frac{7}{2} )}{9 - 4 ( x - \frac{7}{2} ) ^{2} + 9} : 4 (2 (\frac{( 2 x - 7 ) ^{2}}{72 - x ^{2} + 7 x - 10} + \frac{1}{18} - x^{2} + 7 x - 10) + \frac{7 ( 2 x - 7 )}{36 - x ^{2} + 7 x - 10})

= 4 (2 (\frac{( 2 x - 7 ) ^{2}}{72 - x ^{2} + 7 x - 10} + \frac{1}{18} - x^{2} + 7 x - 10) + \frac{7 ( 2 x - 7 )}{36 - x ^{2} + 7 x - 10})

Füge eine Konstante zur Lösung hinzu

= 4 (2 (\frac{( 2 x - 7 ) ^{2}}{72 - x ^{2} + 7 x - 10} + \frac{1}{18} - x^{2} + 7 x - 10) + \frac{7 ( 2 x - 7 )}{36 - x ^{2} + 7 x - 10}) + C

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