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

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

I-evaluate

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

Mga Hakbang ng Solution

Kalkulahin ang 10 sa power ng -2 at kunin ang \frac{1}{100}.

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

Gamitin ang binomial theorem na \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para palawakin ang \left(3-7x\right)^{2}.

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

Gamitin ang binomial theorem na \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para palawakin ang \left(91+292x\right)^{2}.

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

Gamitin ang distributive property para i-multiply ang \frac{1}{100} gamit ang 9-42x+49x^{2}.

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

Gamitin ang distributive property para i-multiply ang \frac{9}{100}-\frac{21}{50}x+\frac{49}{100}x^{2} sa 8281+53144x+85264x^{2} at para pagsamahin ang magkakatulad na term.

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

I-integrate ang sum sa bawat term.

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

I-factor out ang constant sa bawat mga term.

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

Hanapin ang integral ng \frac{74529}{100} gamit ang alituntunin ng talahanayan ng mga karaniwang integral \int a\mathrm{d}x=ax.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x\mathrm{d}x ng \frac{x^{2}}{2}. I-multiply ang \frac{65247}{50} times \frac{x^{2}}{2}.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x^{2}\mathrm{d}x ng \frac{x^{3}}{3}. I-multiply ang -\frac{1058903}{100} times \frac{x^{3}}{3}.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x^{3}\mathrm{d}x ng \frac{x^{4}}{4}. I-multiply ang -\frac{244258}{25} times \frac{x^{4}}{4}.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x^{4}\mathrm{d}x ng \frac{x^{5}}{5}. I-multiply ang \frac{1044484}{25} times \frac{x^{5}}{5}.

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

Kung ang F\left(x\right) ay isang antiderivative ng f\left(x\right), kung gayon ang hanay ng lahat ng antiderivatives ng f\left(x\right) ay ibinibigay ng F\left(x\right)+C. Kaya naman, idagdag ang constant ng integration C\in \mathrm{R} sa resulta.

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

I-differentiate ang w.r.t. x

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