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Beliebt Rechnen >

integral of (t^2(t^2+6))/((t^2+2)^2)

Lösung

\int \frac{t ^{2} ( t ^{2} + 6 )}{( t ^{2} + 2 ) ^{2}} d t

Lösung

t - \frac{1}{2} sin (2 arctan (\frac{t}{2})) + C

Schritte zur Lösung

\int \frac{t ^{2} ( t ^{2} + 6 )}{( t ^{2} + 2 ) ^{2}} d t

Ermittle den Partialbruch von

\frac{t ^{2} ( t ^{2} + 6 )}{( t ^{2} + 2 ) ^{2}} : 1 + \frac{2}{t ^{2} + 2} - \frac{8}{( t ^{2} + 2 ) ^{2}}

= \int 1 + \frac{2}{t ^{2} + 2} - \frac{8}{( t ^{2} + 2 ) ^{2}} d t

Wende die Summenregel an:

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

= \int 1 d t + \int \frac{2}{t ^{2} + 2} d t - \int \frac{8}{( t ^{2} + 2 ) ^{2}} d t

\int 1 d t = t

\int \frac{2}{t ^{2} + 2} d t = 2 arctan (\frac{t}{2})

\int \frac{8}{( t ^{2} + 2 ) ^{2}} d t = 2 (arctan (\frac{t}{2}) + \frac{1}{2} sin (2 arctan (\frac{t}{2})))

= t + 2 arctan (\frac{t}{2}) - 2 (arctan (\frac{t}{2}) + \frac{1}{2} sin (2 arctan (\frac{t}{2})))

Vereinfache

t + 2 arctan (\frac{t}{2}) - 2 (arctan (\frac{t}{2}) + \frac{1}{2} sin (2 arctan (\frac{t}{2}))) : t - \frac{1}{2} sin (2 arctan (\frac{t}{2}))

= t - \frac{1}{2} sin (2 arctan (\frac{t}{2}))

Füge eine Konstante zur Lösung hinzu

= t - \frac{1}{2} sin (2 arctan (\frac{t}{2})) + C

Graph

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