Issolvi ∫ (2x+7)^3 wrt x

$$\int ( 2 x + 7 ) ^ { 3 } d x$$

Evalwa

$$\frac{\left(2x+7\right)^{4}}{8}+С$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} biex tespandi \left(2x+7\right)^{3}.

$$\int 8x^{3}+84x^{2}+294x+343\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 8x^{3}\mathrm{d}x+\int 84x^{2}\mathrm{d}x+\int 294x\mathrm{d}x+\int 343\mathrm{d}x$$

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

$$8\int x^{3}\mathrm{d}x+84\int x^{2}\mathrm{d}x+294\int x\mathrm{d}x+\int 343\mathrm{d}x$$

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

$$2x^{4}+84\int x^{2}\mathrm{d}x+294\int x\mathrm{d}x+\int 343\mathrm{d}x$$

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

$$2x^{4}+28x^{3}+294\int x\mathrm{d}x+\int 343\mathrm{d}x$$

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

$$2x^{4}+28x^{3}+147x^{2}+\int 343\mathrm{d}x$$

Sib l-integru ta' 343 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$2x^{4}+28x^{3}+147x^{2}+343x$$

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.

$$2x^{4}+28x^{3}+147x^{2}+343x+С$$

Iddifferenzja w.r.t. x

$$\left(2x+7\right)^{3}$$