Mathematics

Solve :
$$\displaystyle \int { \cfrac { x-1 }{ \sqrt { x+4 }  }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{x-1}{\sqrt{x+4}}dx}$$

$$=\displaystyle\int{\dfrac{x+4-5}{\sqrt{x+4}}dx}$$

$$=\displaystyle\int{\dfrac{x+4}{\sqrt{x+4}}dx}-5\displaystyle\int{\dfrac{dx}{\sqrt{x+4}}}$$

$$=\displaystyle\int{{\left(x+4\right)}^{1-\frac{1}{2}}}-5\displaystyle\int{{\left(x+4\right)}^{\frac{-1}{2}}dx}$$

$$=\displaystyle\int{{\left(x+4\right)}^{\frac{1}{2}}}-5\displaystyle\int{{\left(x+4\right)}^{\frac{-1}{2}}dx}$$

We know that $$\displaystyle\int{{\left(ax+b\right)}^{n}}=\dfrac{1}{a\left(n+1\right)}{\left(ax+b\right)}^{n+1}$$ 

$$=\dfrac{{\left(x+4\right)}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}-5\dfrac{{\left(x+4\right)}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$$

$$=\dfrac{{\left(x+4\right)}^{\frac{3}{2}}}{\dfrac{3}{2}}-5\dfrac{{\left(x+4\right)}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$$

$$=\dfrac{2}{3}{\left(x+4\right)}^{\frac{3}{2}}-10{\left(x+4\right)}^{\frac{1}{2}}+c$$
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Subjective Medium Published on 17th 09, 2020
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