integral of (x+1)/(sqrt(2x-1))

Lösung

\int \frac{x + 1}{2 x - 1} d x

\frac{1}{3} (x + 1) 2 x - 1 + 2 x - 1 + C

Schritte zur Lösung

\int \frac{x + 1}{2 x - 1} d x

Wende U-Substitution an

= \int \frac{u ^{2} + 1}{2} + 1 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

= \int \frac{u ^{2} + 1}{2} d u + \int 1 d u

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

\int 1 d u = u

= \frac{1}{2} (\frac{u ^{3}}{3} + u) + u

Setze in

u = 2 x - 1

ein

= \frac{1}{2} (\frac{( 2 x - 1 ) ^{3}}{3} + 2 x - 1) + 2 x - 1

Vereinfache

\frac{1}{2} (\frac{( 2 x - 1 ) ^{3}}{3} + 2 x - 1) + 2 x - 1 : \frac{1}{3} (x + 1) 2 x - 1 + 2 x - 1

= \frac{1}{3} (x + 1) 2 x - 1 + 2 x - 1

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (x + 1) 2 x - 1 + 2 x - 1 + C

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