Issolvi ∫ (4x^7+4x+4)(28x^6+4) wrt x

$$\int ( 4 x ^ { 7 } + 4 x + 4 ) ( 28 x ^ { 6 } + 4 ) d x$$

Evalwa

$$8x^{14}+16x^{8}+16x^{7}+8x^{2}+16x+С$$

Passi tas-Soluzzjonijiet

Uża l-propjetà distributtiva biex timmultiplika 4x^{7}+4x+4 b'28x^{6}+4 u kkombina termini simili.

$$\int 112x^{13}+128x^{7}+16x+112x^{6}+16\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 112x^{13}\mathrm{d}x+\int 128x^{7}\mathrm{d}x+\int 16x\mathrm{d}x+\int 112x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

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

$$112\int x^{13}\mathrm{d}x+128\int x^{7}\mathrm{d}x+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{13}\mathrm{d}x ma' \frac{x^{14}}{14}. Immultiplika 112 b'\frac{x^{14}}{14}.

$$8x^{14}+128\int x^{7}\mathrm{d}x+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{7}\mathrm{d}x ma' \frac{x^{8}}{8}. Immultiplika 128 b'\frac{x^{8}}{8}.

$$8x^{14}+16x^{8}+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

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

$$8x^{14}+16x^{8}+8x^{2}+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

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 112 b'\frac{x^{7}}{7}.

$$8x^{14}+16x^{8}+8x^{2}+16x^{7}+\int 16\mathrm{d}x$$

Sib l-integru ta' 16 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$8x^{14}+16x^{8}+8x^{2}+16x^{7}+16x$$

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.

$$8x^{14}+16x^{8}+16x^{7}+8x^{2}+16x+С$$

Iddifferenzja w.r.t. x

$$16\left(7x^{6}+1\right)\left(x^{7}+x+1\right)$$