Réitigh ∫ (5x+2)^3 wrt x=

$$\int ( 5 x + 2 ) ^ { 3 } d x =$$

Luacháil

$$\frac{\left(5x+2\right)^{4}}{20}+С$$

Céimeanna Réitigh

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} chun \left(5x+2\right)^{3} a leathnú.

$$\int 125x^{3}+150x^{2}+60x+8\mathrm{d}x$$

Measc an tsuim téarma fá téarma.

$$\int 125x^{3}\mathrm{d}x+\int 150x^{2}\mathrm{d}x+\int 60x\mathrm{d}x+\int 8\mathrm{d}x$$

Fág an leanúnach sna téarmaí as an áireamh.

$$125\int x^{3}\mathrm{d}x+150\int x^{2}\mathrm{d}x+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{3}\mathrm{d}x le \frac{x^{4}}{4}. Méadaigh 125 faoi \frac{x^{4}}{4}.

$$\frac{125x^{4}}{4}+150\int x^{2}\mathrm{d}x+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{2}\mathrm{d}x le \frac{x^{3}}{3}. Méadaigh 150 faoi \frac{x^{3}}{3}.

$$\frac{125x^{4}}{4}+50x^{3}+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x\mathrm{d}x le \frac{x^{2}}{2}. Méadaigh 60 faoi \frac{x^{2}}{2}.

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+\int 8\mathrm{d}x$$

Aimsigh suimeálaithe do 8 ag baint úsáid as an tábla do suimeálaithe coitianta riail\int a\mathrm{d}x=ax.

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+8x$$

Má tá F\left(x\right) mar frithdhíorthach do f\left(x\right), beidh tacar do frithdhíorthach uile do f\left(x\right) a thabhairt ag F\left(x\right)+C. Mar sin de, cur an comhtháthú leanúnach C\in \mathrm{R} don toradh.

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+8x+С$$

Difreálaigh w.r.t. x

$$\left(5x+2\right)^{3}$$