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Beliebt Rechnen >

integral of (8t^3+10)/(4t^2-4t+5)

Lösung

\int \frac{8 t ^{3} + 10}{4 t ^{2} - 4 t + 5} d t

Lösung

t^{2} + 2 t - \frac{1}{4} ln 4 t^{2} - 4 t + 5 - \frac{1}{4} arctan (t - \frac{1}{2}) + C

Schritte zur Lösung

\int \frac{8 t ^{3} + 10}{4 t ^{2} - 4 t + 5} d t

Lange Division

\frac{8 t ^{3} + 10}{4 t ^{2} - 4 t + 5} : 2 t + 2 - \frac{2 t}{4 t ^{2} - 4 t + 5}

= \int 2 t + 2 - \frac{2 t}{4 t ^{2} - 4 t + 5} 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 2 t d t + \int 2 d t - \int \frac{2 t}{4 t ^{2} - 4 t + 5} d t

\int 2 t d t = t^{2}

\int 2 d t = 2 t

\int \frac{2 t}{4 t ^{2} - 4 t + 5} d t = \frac{1}{4} ln 4 t^{2} - 4 t + 5 + \frac{1}{4} arctan (t - \frac{1}{2})

= t^{2} + 2 t - (\frac{1}{4} ln 4 t^{2} - 4 t + 5 + \frac{1}{4} arctan (t - \frac{1}{2}))

Vereinfache

= t^{2} + 2 t - \frac{1}{4} ln 4 t^{2} - 4 t + 5 - \frac{1}{4} arctan (t - \frac{1}{2})

Füge eine Konstante zur Lösung hinzu

= t^{2} + 2 t - \frac{1}{4} ln 4 t^{2} - 4 t + 5 - \frac{1}{4} arctan (t - \frac{1}{2}) + C

Graph

Beliebte Beispiele

integral of x^4+1

\int x^{4} + 1 d x

integral of (3x+2)/(sqrt(19-5x+x^2))

\int \frac{3 x + 2}{19 - 5 x + x ^{2}} d x

limit as x approaches c of 2.5(-2)

x \to c lim (2.5 (- 2))

derivative Q(x)=x^4+8x^2+16

d er i v a t i v e Q (x) = x^{4} + 8 x^{2} + 16

integral of x^2e^{1-x}

\int x^{2} e^{1 - x} d x