Issolvi ∫ x(x+5)^3 wrt x

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

Evalwa

$$\frac{x^{5}}{5}+\frac{15x^{4}}{4}+25x^{3}+\frac{125x^{2}}{2}+С$$

Passi tas-Soluzzjonijiet

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

$$\int x\left(x^{3}+15x^{2}+75x+125\right)\mathrm{d}x$$

Uża l-propjetà distributtiva biex timmultiplika x b'x^{3}+15x^{2}+75x+125.

$$\int x^{4}+15x^{3}+75x^{2}+125x\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int x^{4}\mathrm{d}x+\int 15x^{3}\mathrm{d}x+\int 75x^{2}\mathrm{d}x+\int 125x\mathrm{d}x$$

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

$$\int x^{4}\mathrm{d}x+15\int x^{3}\mathrm{d}x+75\int x^{2}\mathrm{d}x+125\int x\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{4}\mathrm{d}x ma' \frac{x^{5}}{5}.

$$\frac{x^{5}}{5}+15\int x^{3}\mathrm{d}x+75\int x^{2}\mathrm{d}x+125\int x\mathrm{d}x$$

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 15 b'\frac{x^{4}}{4}.

$$\frac{x^{5}}{5}+\frac{15x^{4}}{4}+75\int x^{2}\mathrm{d}x+125\int x\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{2}\mathrm{d}x ma' \frac{x^{3}}{3}. Immultiplika 75 b'\frac{x^{3}}{3}.

$$\frac{x^{5}}{5}+\frac{15x^{4}}{4}+25x^{3}+125\int x\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x\mathrm{d}x ma' \frac{x^{2}}{2}. Immultiplika 125 b'\frac{x^{2}}{2}.

$$\frac{x^{5}}{5}+\frac{15x^{4}}{4}+25x^{3}+\frac{125x^{2}}{2}$$

Issimplifika.

$$\frac{125x^{2}}{2}+25x^{3}+\frac{15x^{4}}{4}+\frac{x^{5}}{5}$$

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.

$$\frac{125x^{2}}{2}+25x^{3}+\frac{15x^{4}}{4}+\frac{x^{5}}{5}+С$$

Iddifferenzja w.r.t. x

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