Datrys ∫ 10^-2(3-7x)^2(91+292x)^2 wrt x

$$\int 10 ^ { - 2 } ( 3 - 7 x ) ^ { 2 } ( 91 + 292 x ) ^ { 2 } d x$$

Enrhifo

$$\frac{1044484x^{5}}{125}-\frac{122129x^{4}}{50}-\frac{1058903x^{3}}{300}+\frac{65247x^{2}}{100}+\frac{74529x}{100}+С$$

Camau Ateb

Cyfrifo 10 i bŵer -2 a chael \frac{1}{100}.

$$\int \frac{1}{100}\left(3-7x\right)^{2}\left(91+292x\right)^{2}\mathrm{d}x$$

Defnyddio'r theorem binomaidd \left(a-b\right)^{2}=a^{2}-2ab+b^{2} i ehangu'r \left(3-7x\right)^{2}.

$$\int \frac{1}{100}\left(9-42x+49x^{2}\right)\left(91+292x\right)^{2}\mathrm{d}x$$

Defnyddio'r theorem binomaidd \left(a+b\right)^{2}=a^{2}+2ab+b^{2} i ehangu'r \left(91+292x\right)^{2}.

$$\int \frac{1}{100}\left(9-42x+49x^{2}\right)\left(8281+53144x+85264x^{2}\right)\mathrm{d}x$$

Defnyddio’r briodwedd ddosbarthu i luosi \frac{1}{100} â 9-42x+49x^{2}.

$$\int \left(\frac{9}{100}-\frac{21}{50}x+\frac{49}{100}x^{2}\right)\left(8281+53144x+85264x^{2}\right)\mathrm{d}x$$

Defnyddio’r briodwedd ddosbarthu i luosi \frac{9}{100}-\frac{21}{50}x+\frac{49}{100}x^{2} â 8281+53144x+85264x^{2} a chyfuno termau tebyg.

$$\int \frac{74529}{100}+\frac{65247}{50}x-\frac{1058903}{100}x^{2}-\frac{244258}{25}x^{3}+\frac{1044484}{25}x^{4}\mathrm{d}x$$

Integreiddio'r swm fesul term.

$$\int \frac{74529}{100}\mathrm{d}x+\int \frac{65247x}{50}\mathrm{d}x+\int -\frac{1058903x^{2}}{100}\mathrm{d}x+\int -\frac{244258x^{3}}{25}\mathrm{d}x+\int \frac{1044484x^{4}}{25}\mathrm{d}x$$

Ffactoreiddio allan y cysonyn ym mhob un o'r termau.

$$\int \frac{74529}{100}\mathrm{d}x+\frac{65247\int x\mathrm{d}x}{50}-\frac{1058903\int x^{2}\mathrm{d}x}{100}-\frac{244258\int x^{3}\mathrm{d}x}{25}+\frac{1044484\int x^{4}\mathrm{d}x}{25}$$

Canfod integryn \frac{74529}{100} gan ddefnyddio'r rheol tabl o integrynnau cyffredin \int a\mathrm{d}x=ax.

$$\frac{74529x}{100}+\frac{65247\int x\mathrm{d}x}{50}-\frac{1058903\int x^{2}\mathrm{d}x}{100}-\frac{244258\int x^{3}\mathrm{d}x}{25}+\frac{1044484\int x^{4}\mathrm{d}x}{25}$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x\mathrm{d}x gyda \frac{x^{2}}{2}. Lluoswch \frac{65247}{50} â \frac{x^{2}}{2}.

$$\frac{74529x}{100}+\frac{65247x^{2}}{100}-\frac{1058903\int x^{2}\mathrm{d}x}{100}-\frac{244258\int x^{3}\mathrm{d}x}{25}+\frac{1044484\int x^{4}\mathrm{d}x}{25}$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{2}\mathrm{d}x gyda \frac{x^{3}}{3}. Lluoswch -\frac{1058903}{100} â \frac{x^{3}}{3}.

$$\frac{74529x}{100}+\frac{65247x^{2}}{100}-\frac{1058903x^{3}}{300}-\frac{244258\int x^{3}\mathrm{d}x}{25}+\frac{1044484\int x^{4}\mathrm{d}x}{25}$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{3}\mathrm{d}x gyda \frac{x^{4}}{4}. Lluoswch -\frac{244258}{25} â \frac{x^{4}}{4}.

$$\frac{74529x}{100}+\frac{65247x^{2}}{100}-\frac{1058903x^{3}}{300}-\frac{122129x^{4}}{50}+\frac{1044484\int x^{4}\mathrm{d}x}{25}$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{4}\mathrm{d}x gyda \frac{x^{5}}{5}. Lluoswch \frac{1044484}{25} â \frac{x^{5}}{5}.

$$\frac{74529x}{100}+\frac{65247x^{2}}{100}-\frac{1058903x^{3}}{300}-\frac{122129x^{4}}{50}+\frac{1044484x^{5}}{125}$$

Os yw F\left(x\right) yn integryn amhendant o f\left(x\right), yna bydd F\left(x\right)+C yn rhoi’r set o holl integrynnau amhendant f\left(x\right). Felly, ychwanegwch gysonyn yr integryn C\in \mathrm{R} at y canlyniad.

$$\frac{74529x}{100}+\frac{65247x^{2}}{100}-\frac{1058903x^{3}}{300}-\frac{122129x^{4}}{50}+\frac{1044484x^{5}}{125}+С$$

Gwahaniaethu w.r.t. x

$$\frac{\left(\left(7x-3\right)\left(292x+91\right)\right)^{2}}{100}$$