Rešite ∫ (from - 5 to - 1) of 7/3x^4(6-x^5)^3 wrt x=

$$\int _ { - 5 } ^ { - 1 } \frac { 7 } { 3 } x ^ { 4 } ( 6 - x ^ { 5 } ) ^ { 3 } d x =$$

Ovrednoti

$$11211895985444$$

Koraki rešitve

Uporabite binomski izrek \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}, da razširite \left(6-x^{5}\right)^{3}.

$$\int _{-5}^{-1}\frac{7}{3}x^{4}\left(216-108x^{5}+18\left(x^{5}\right)^{2}-\left(x^{5}\right)^{3}\right)\mathrm{d}x$$

Če želite potenco potencirati z drugo potenco, pomnožite eksponente. Pomnožite 5 in 2, da dobite 10.

$$\int _{-5}^{-1}\frac{7}{3}x^{4}\left(216-108x^{5}+18x^{10}-\left(x^{5}\right)^{3}\right)\mathrm{d}x$$

Če želite potenco potencirati z drugo potenco, pomnožite eksponente. Pomnožite 5 in 3, da dobite 15.

$$\int _{-5}^{-1}\frac{7}{3}x^{4}\left(216-108x^{5}+18x^{10}-x^{15}\right)\mathrm{d}x$$

Uporabite distributivnost, da pomnožite \frac{7}{3}x^{4} s/z 216-108x^{5}+18x^{10}-x^{15}.

$$\int _{-5}^{-1}504x^{4}-252x^{9}+42x^{14}-\frac{7}{3}x^{19}\mathrm{d}x$$

Najprej ovrednotite nedoločni integral.

$$\int 504x^{4}-252x^{9}+42x^{14}-\frac{7x^{19}}{3}\mathrm{d}x$$

Vsoto povežite z izrazom.

$$\int 504x^{4}\mathrm{d}x+\int -252x^{9}\mathrm{d}x+\int 42x^{14}\mathrm{d}x+\int -\frac{7x^{19}}{3}\mathrm{d}x$$

Faktorizirajte konstanto v vseh izrazih.

$$504\int x^{4}\mathrm{d}x-252\int x^{9}\mathrm{d}x+42\int x^{14}\mathrm{d}x-\frac{7\int x^{19}\mathrm{d}x}{3}$$

Ker \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} za k\neq -1, zamenjajte \int x^{4}\mathrm{d}x s \frac{x^{5}}{5}. Pomnožite 504 s/z \frac{x^{5}}{5}.

$$\frac{504x^{5}}{5}-252\int x^{9}\mathrm{d}x+42\int x^{14}\mathrm{d}x-\frac{7\int x^{19}\mathrm{d}x}{3}$$

Ker \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} za k\neq -1, zamenjajte \int x^{9}\mathrm{d}x s \frac{x^{10}}{10}. Pomnožite -252 s/z \frac{x^{10}}{10}.

$$\frac{504x^{5}}{5}-\frac{126x^{10}}{5}+42\int x^{14}\mathrm{d}x-\frac{7\int x^{19}\mathrm{d}x}{3}$$

Ker \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} za k\neq -1, zamenjajte \int x^{14}\mathrm{d}x s \frac{x^{15}}{15}. Pomnožite 42 s/z \frac{x^{15}}{15}.

$$\frac{504x^{5}}{5}-\frac{126x^{10}}{5}+\frac{14x^{15}}{5}-\frac{7\int x^{19}\mathrm{d}x}{3}$$

Ker \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} za k\neq -1, zamenjajte \int x^{19}\mathrm{d}x s \frac{x^{20}}{20}. Pomnožite -\frac{7}{3} s/z \frac{x^{20}}{20}.

$$\frac{504x^{5}}{5}-\frac{126x^{10}}{5}+\frac{14x^{15}}{5}-\frac{7x^{20}}{60}$$

Določen integral je integral izraza, ovrednotenega pri zgornji omejitvi integriranja, minus integral, ovrednoten pri spodnji omejitvi integriranja.

$$\frac{504}{5}\left(-1\right)^{5}-\frac{126}{5}\left(-1\right)^{10}+\frac{14}{5}\left(-1\right)^{15}-\frac{7}{60}\left(-1\right)^{20}-\left(\frac{504}{5}\left(-5\right)^{5}-\frac{126}{5}\left(-5\right)^{10}+\frac{14}{5}\left(-5\right)^{15}-\frac{7}{60}\left(-5\right)^{20}\right)$$

Poenostavite.

$$11211895985444$$