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Beliebt Rechnen >

integral of (y^3)/((y^2-4)^{1/2)}

Lösung

\int \frac{y ^{3}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} d y

Lösung

\frac{y ^{4}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - \frac{2}{3} (y^{2} - 4)^{\frac{3}{2}} - \frac{16}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - 4 (y^{2} - 4)^{\frac{1}{2}} + C

Schritte zur Lösung

\int \frac{y ^{3}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} d y

Wende die partielle Integration an

= \frac{y ^{4}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - \int \frac{y ( 2 y ^{4} - 12 y ^{2} )}{( y ^{2} - 4 ) ^{\frac{3}{2}}} d y

\int \frac{y ( 2 y ^{4} - 12 y ^{2} )}{( y ^{2} - 4 ) ^{\frac{3}{2}}} d y = \frac{2}{3} (y^{2} - 4)^{\frac{3}{2}} + \frac{16}{( y ^{2} - 4 ) ^{\frac{1}{2}}} + 4 (y^{2} - 4)^{\frac{1}{2}}

= \frac{y ^{4}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - (\frac{2}{3} (y^{2} - 4)^{\frac{3}{2}} + \frac{16}{( y ^{2} - 4 ) ^{\frac{1}{2}}} + 4 (y^{2} - 4)^{\frac{1}{2}})

Vereinfache

= \frac{y ^{4}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - \frac{2}{3} (y^{2} - 4)^{\frac{3}{2}} - \frac{16}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - 4 (y^{2} - 4)^{\frac{1}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{y ^{4}}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - \frac{2}{3} (y^{2} - 4)^{\frac{3}{2}} - \frac{16}{( y ^{2} - 4 ) ^{\frac{1}{2}}} - 4 (y^{2} - 4)^{\frac{1}{2}} + C

Graph

Beliebte Beispiele

integral of (e^x-3)

\int (e^{x} - 3) d x

integral of (2t-t^2)sin(t^2-(t^3)/3)

\int (2 t - t^{2}) sin (t^{2} - \frac{t ^{3}}{3}) d t

integral of ln(1-4x)

\int ln (1 - 4 x) d x

integral of (x^2+4x+1)/((x^2-1)(x+3))

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

integral of (x^2)/((x^2+1)^3)

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