integral of 3χ^5sqrt(χ^3+1)

Lösung

\int 3 χ^{5} χ^{3} + 1 d χ

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

Schritte zur Lösung

\int 3 χ^{5} χ^{3} + 1 d χ

Entferne die Konstante:

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

= 3 \cdot \int χ^{5} χ^{3} + 1 d χ

Wende U-Substitution an

= 3 \cdot \int \frac{( u - 1 ) u}{3} d u

Entferne die Konstante:

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

= 3 \cdot \frac{1}{3} \cdot \int (u - 1) u d u

Multipliziere aus

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

= 3 \cdot \frac{1}{3} \cdot \int u^{\frac{3}{2}} - 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

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

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

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

= 3 \cdot \frac{1}{3} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = χ^{3} + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\int e^{2 x} (1 - 6 e^{2 x})^{2} d x

d er i v a t i v e z

d er i v a t i v e x (x^{2} + 6) (x + 9)

t an g e n t f (x) = e^{4 x} cos (π x), at x = 0

t a y l or cos (3 x)