Issolvi ∫ (from 0 to 1) of 4x(1+x)^3 wrt x

$$\int _ { 0 } ^ { 1 } 4 x ( 1 + x ) ^ { 3 } d x$$

Evalwa

$$\frac{49}{5}=9.8$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} biex tespandi \left(1+x\right)^{3}.

$$\int _{0}^{1}4x\left(1+3x+3x^{2}+x^{3}\right)\mathrm{d}x$$

Uża l-propjetà distributtiva biex timmultiplika 4x b'1+3x+3x^{2}+x^{3}.

$$\int _{0}^{1}4x+12x^{2}+12x^{3}+4x^{4}\mathrm{d}x$$

L-ewwel evalwa l-integru indefinit.

$$\int 4x+12x^{2}+12x^{3}+4x^{4}\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 4x\mathrm{d}x+\int 12x^{2}\mathrm{d}x+\int 12x^{3}\mathrm{d}x+\int 4x^{4}\mathrm{d}x$$

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

$$4\int x\mathrm{d}x+12\int x^{2}\mathrm{d}x+12\int x^{3}\mathrm{d}x+4\int x^{4}\mathrm{d}x$$

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 4 b'\frac{x^{2}}{2}.

$$2x^{2}+12\int x^{2}\mathrm{d}x+12\int x^{3}\mathrm{d}x+4\int x^{4}\mathrm{d}x$$

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 12 b'\frac{x^{3}}{3}.

$$2x^{2}+4x^{3}+12\int x^{3}\mathrm{d}x+4\int x^{4}\mathrm{d}x$$

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

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

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}. Immultiplika 4 b'\frac{x^{5}}{5}.

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

L-integru definit huwa l-antiderivattiv tal-espressjoni evalwata fil-limitu superjuri tal-integrazzjoni minus l-antiderivattiv evalwat fil-limitu inferjuri tal-integrazzjoni.

$$2\times 1^{2}+4\times 1^{3}+3\times 1^{4}+\frac{4}{5}\times 1^{5}-\left(2\times 0^{2}+4\times 0^{3}+3\times 0^{4}+\frac{4}{5}\times 0^{5}\right)$$

Issimplifika.

$$\frac{49}{5}$$