Mathematics

Evaluate: $$\displaystyle\int_{4}^{9}\dfrac{\sqrt{x}}{(30-x^{3/2})^{2}}\, dx$$


SOLUTION
$$\displaystyle \int_4^9 \dfrac{\sqrt{x}}{(30-x^{3/2})^2} dx$$

Let $$t = 30 - x^{3/2}$$

$$dt = 0 - \dfrac{3}{2} x^{1/2} . dx = -\dfrac{3}{2} \sqrt{x} dx$$

$$\dfrac{-2}{3} dt = \sqrt{x} dx$$

$$\Rightarrow \displaystyle \int_4^9 \dfrac{-2/3. dt}{t^2} = \dfrac{-2}{3} \int_4^9 \dfrac{dt}{t^2} = \dfrac{-2}{3} \left[\dfrac{-1}{t} \right]_4^9$$

$$= \dfrac{-2}{3} \left[\dfrac{-1}{30-x^{3/2}} \right]_4^9 = \dfrac{+2}{3} \left[\dfrac{1}{30-27} - \dfrac{1}{30-3} \right]$$

$$= \dfrac{2}{3} \left[\dfrac{1}{3} - \dfrac{1}{22} \right] = \dfrac{2}{3} \times \dfrac{19}{66} = \dfrac{19}{99}$$
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Subjective Medium Published on 17th 09, 2020
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