Mathematics

Evaluate the following integral:
$$\int { \cfrac { 1 }{ 1+\sqrt { x }  }  } dx$$


SOLUTION
$$\displaystyle \int \dfrac{1}{1+\sqrt{x}}dx$$

Multiplying with root $$x$$

$$\displaystyle \int\dfrac{1}{\sqrt{x}} \dfrac{\sqrt{x}}{1+\sqrt{x}}dx$$

$$\displaystyle \int\dfrac{1}{\sqrt{x}} \left[1- \dfrac{1}{1+\sqrt{x}}\right]dx$$

$$\displaystyle \int \left[\dfrac{1}{\sqrt{x}} - \dfrac{1}{\sqrt{x}(1+\sqrt{x})}\right]dx$$
                                $$\downarrow$$
                          $$1+\sqrt{x}=t$$
                          $$\dfrac{1}{2\sqrt{x}}dx=dt$$

$$\dfrac{x^{\dfrac{-1}{2}+1}}{-\dfrac{1}{2}+1}-\displaystyle\int\dfrac{2dt}{t}$$

$$2\sqrt{x}-2\log t$$

$$2(\sqrt{x}-\log(1+\sqrt{x}))+c$$
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Subjective Medium Published on 17th 09, 2020
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