Issolvi ∫ (sqrt{x}+sqrt[3]{x^4}+3sqrt{x^3}) wrt x

$$\int ( \sqrt { x } + \sqrt[ 3 ] { x ^ { 4 } } + 3 \sqrt { x ^ { 3 } } ) d x$$

Evalwa

$$\frac{6x^{\frac{5}{2}}}{5}+\frac{3x^{\frac{7}{3}}}{7}+\frac{2x^{\frac{3}{2}}}{3}+С$$

Passi tas-Soluzzjonijiet

Tintegra s-somma terminu b'terminu.

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

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

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

Erġa' ikteb \sqrt{x} bħala x^{\frac{1}{2}}. Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{\frac{1}{2}}\mathrm{d}x ma' \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Issimplifika.

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

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{\frac{4}{3}}\mathrm{d}x ma' \frac{3x^{\frac{7}{3}}}{7}.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{3x^{\frac{7}{3}}}{7}+3\int x^{\frac{3}{2}}\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{\frac{3}{2}}\mathrm{d}x ma' \frac{2x^{\frac{5}{2}}}{5}. Immultiplika 3 b'\frac{2x^{\frac{5}{2}}}{5}.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{3x^{\frac{7}{3}}}{7}+\frac{6x^{\frac{5}{2}}}{5}$$

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.

$$\frac{2x^{\frac{3}{2}}}{3}+\frac{3x^{\frac{7}{3}}}{7}+\frac{6x^{\frac{5}{2}}}{5}+С$$

Iddifferenzja w.r.t. x

$$\sqrt{x}\left(3x+x^{\frac{5}{6}}+1\right)$$