integral of (t+2)sqrt(2+3t)

Lösung

\int (t + 2) 2 + 3 t d t

\frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int (t + 2) 2 + 3 t d t

Wende U-Substitution an

= \int \frac{u ( u + 4 )}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int u (u + 4) d u

Multipliziere aus

u (u + 4) : u^{\frac{3}{2}} + 4 u

= \frac{1}{9} \cdot \int u^{\frac{3}{2}} + 4 u 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

= \frac{1}{9} (\int u^{\frac{3}{2}} d u + \int 4 u d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 4 u d u = \frac{8}{3} u^{\frac{3}{2}}

= \frac{1}{9} (\frac{2}{5} u^{\frac{5}{2}} + \frac{8}{3} u^{\frac{3}{2}})

Setze in

u = 2 + 3 t

ein

= \frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9} (\frac{2}{5} (2 + 3 t)^{\frac{5}{2}} + \frac{8}{3} (2 + 3 t)^{\frac{3}{2}}) + C

\int (cos (x)) (e^{s i n (x)}) d x

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

\int \frac{x cos ( x ^{2} )}{sin ^{2} ( x ^{2} )} d x

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

\int \frac{1}{y} + y d y