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G(x)=sqrt(3x) g(x)=G^(')(x) int_(3)^(12)g(x)dx=

G(x)=3x G(x)=\sqrt{3 x} \newlineg(x)=G′(x) g(x)=G^{\prime}(x) \newline∫312g(x)dx= \int_{3}^{12} g(x) d x=

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Q. G(x)=3x G(x)=\sqrt{3 x} \newlineg(x)=G′(x) g(x)=G^{\prime}(x) \newline∫312g(x)dx= \int_{3}^{12} g(x) d x=
  1. Find Derivative of G(x)G(x): First, find the derivative of G(x)=3xG(x) = \sqrt{3x} to get g(x)g(x).g(x)=G′(x)=(12)(3x)−12×3=(32)(3x)−12g(x) = G'(x) = \left(\frac{1}{2}\right)(3x)^{-\frac{1}{2}} \times 3 = \left(\frac{3}{2}\right)(3x)^{-\frac{1}{2}}
  2. Simplify g(x)g(x): Now, simplify g(x)g(x).g(x) = \left(\frac{\(3\)}{\(2\)}\right)\left(\frac{\(1\)}{\sqrt{\(3\)x}}\right) = \left(\frac{\(3\)}{\(2\)}\right)\left(\frac{\(1\)}{\sqrt{\(3\)}\sqrt{x}}\right) = \left(\frac{\(3\)}{\(2\)}\right)\left(\frac{\(1\)}{\sqrt{\(3\)}x^{\frac{\(1\)}{\(2\)}}}\right) = \frac{\(3\)}{\(2\)\sqrt{\(3\)}\sqrt{x}}
  3. Set Up Integral: Next, set up the integral of \(g(x) from 33 to 1212.∫312g(x) dx=∫312(323x)dx\int_{3}^{12}g(x)\,dx = \int_{3}^{12}\left(\frac{3}{2\sqrt{3}\sqrt{x}}\right)dx
  4. Integrate g(x)g(x): Integrate g(x)g(x) with respect to xx.∫312(323x)dx=(323)⋅∫312x−12dx\int_{3}^{12}\left(\frac{3}{2\sqrt{3}\sqrt{x}}\right)dx = \left(\frac{3}{2\sqrt{3}}\right) \cdot \int_{3}^{12}x^{-\frac{1}{2}}dx
  5. Find Antiderivative: Find the antiderivative of x−12x^{-\frac{1}{2}}. \newline∫x−12 dx=2x12+C\int x^{-\frac{1}{2}}\,dx = 2x^{\frac{1}{2}} + C
  6. Apply Antiderivative: Now, apply the antiderivative to the integral with the limits from 33 to 1212. \newline323\frac{3}{2\sqrt{3}} * [2x12][2x^{\frac{1}{2}}] from 33 to 1212 = 323\frac{3}{2\sqrt{3}} * [2⋅1212−2⋅312][2\cdot12^{\frac{1}{2}} - 2\cdot3^{\frac{1}{2}}]
  7. Simplify Expression: Simplify the expression.\newline(323)×[212−23]=(323)×[24×3−23]=(323)×[2×23−23](\frac{3}{2\sqrt{3}}) \times [2\sqrt{12} - 2\sqrt{3}] = (\frac{3}{2\sqrt{3}}) \times [2\sqrt{4\times3} - 2\sqrt{3}] = (\frac{3}{2\sqrt{3}}) \times [2\times2\sqrt{3} - 2\sqrt{3}]
  8. Further Simplify: Further simplify the expression.\newline(323)×[43−23]=(323)×[23]=32×2=3(\frac{3}{2\sqrt{3}}) \times [4\sqrt{3} - 2\sqrt{3}] = (\frac{3}{2\sqrt{3}}) \times [2\sqrt{3}] = \frac{3}{2} \times 2 = 3

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