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

Lösung

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

2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int 2 (u^{4} - 2 u^{2} + 2) d u

Entferne die Konstante:

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

= 2 \cdot \int u^{4} - 2 u^{2} + 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

= 2 (\int u^{4} d u - \int 2 u^{2} d u + \int 2 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 2 u^{2} d u = \frac{2 u ^{3}}{3}

\int 2 d u = 2 u

= 2 (\frac{u ^{5}}{5} - \frac{2 u ^{3}}{3} + 2 u)

Setze in

u = x + 1

ein

= 2 (\frac{( x + 1 ) ^{5}}{5} - \frac{2 ( x + 1 ) ^{3}}{3} + 2 x + 1)

Vereinfache

2 (\frac{( x + 1 ) ^{5}}{5} - \frac{2 ( x + 1 ) ^{3}}{3} + 2 x + 1) : 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1)

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1)

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} - \frac{2}{3} (x + 1)^{\frac{3}{2}} + 2 x + 1) + C

\int e^{5 x} cos (\frac{x}{3}) d x

in t e g r a l e^{(- x^{2})}

\int \frac{sin ( 2 x )}{y} d x

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

\int h^{\frac{1}{2}} d h