Issolvi ∫ (2x^3+5)^3

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

Evalwa

$$\frac{4x^{10}}{5}+\frac{60x^{7}}{7}+\frac{75x^{4}}{2}+125x+С$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} biex tespandi \left(2x^{3}+5\right)^{3}.

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

Biex tgħolli l-qawwa ta' numru għal qawwa oħra, immultiplika l-esponenti. Immultiplika 3 u 3 biex tikseb 9.

$$\int 8x^{9}+60\left(x^{3}\right)^{2}+150x^{3}+125\mathrm{d}x$$

Biex tgħolli l-qawwa ta' numru għal qawwa oħra, immultiplika l-esponenti. Immultiplika 3 u 2 biex tikseb 6.

$$\int 8x^{9}+60x^{6}+150x^{3}+125\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 8x^{9}\mathrm{d}x+\int 60x^{6}\mathrm{d}x+\int 150x^{3}\mathrm{d}x+\int 125\mathrm{d}x$$

Iffattura ‘l barra l-kostanti f’kull wieħed minn dawn it-termini.

$$8\int x^{9}\mathrm{d}x+60\int x^{6}\mathrm{d}x+150\int x^{3}\mathrm{d}x+\int 125\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{9}\mathrm{d}x ma' \frac{x^{10}}{10}. Immultiplika 8 b'\frac{x^{10}}{10}.

$$\frac{4x^{10}}{5}+60\int x^{6}\mathrm{d}x+150\int x^{3}\mathrm{d}x+\int 125\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{6}\mathrm{d}x ma' \frac{x^{7}}{7}. Immultiplika 60 b'\frac{x^{7}}{7}.

$$\frac{4x^{10}}{5}+\frac{60x^{7}}{7}+150\int x^{3}\mathrm{d}x+\int 125\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{3}\mathrm{d}x ma' \frac{x^{4}}{4}. Immultiplika 150 b'\frac{x^{4}}{4}.

$$\frac{4x^{10}}{5}+\frac{60x^{7}}{7}+\frac{75x^{4}}{2}+\int 125\mathrm{d}x$$

Sib l-integru ta' 125 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$\frac{4x^{10}}{5}+\frac{60x^{7}}{7}+\frac{75x^{4}}{2}+125x$$

Issimplifika.

$$125x+\frac{75x^{4}}{2}+\frac{60x^{7}}{7}+\frac{4x^{10}}{5}$$

Jekk F\left(x\right) huwa antiderivattiv ta' f\left(x\right), allura s-sett tal-antiderivati kollha ta' f\left(x\right) jingħata minn F\left(x\right)+C. Għalhekk, żid il-kostanti ta' integrazzjoni C\in \mathrm{R} mar-riżultat.

$$125x+\frac{75x^{4}}{2}+\frac{60x^{7}}{7}+\frac{4x^{10}}{5}+С$$

Iddifferenzja w.r.t. x

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