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Question

Solve :
$$\displaystyle \int { \cfrac { 1 }{ { \left( 7x-5 \right)  }^{ 3 } } +\cfrac { 1 }{ \sqrt { 5x-4 }  }  } dx\quad \quad $$


Solution

Given $$\displaystyle \int { \cfrac { 1 }{ { \left( 7x-5 \right)  }^{ 3 } } +\cfrac { 1 }{ \sqrt { 5x-4 }  }  } dx\quad \quad $$

$$=\displaystyle\int{{\left(7x-5\right)}^{-3}}+\displaystyle\int{{\left(5x-4\right)}^{\frac{-1}{2}}dx}$$

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

$$=\dfrac{1}{7\left(-3+1\right)}{\left(7x-5\right)}^{-3+1}+\dfrac{1}{5\left(\dfrac{-1}{2}+1\right)}{\left(5x-4\right)}^{\frac{-1}{2}+1}+C$$

$$=\dfrac{1}{7\left(-2\right)}{\left(7x-5\right)}^{-2}+\dfrac{1}{5\left(\dfrac{-1+2}{2}\right)}{\left(5x-4\right)}^{\frac{-1+2}{2}}+C$$

$$=\dfrac{1}{-14}{\left(7x-5\right)}^{-2}+\dfrac{2}{5}{\left(5x-4\right)}^{\frac{1}{2}}+C$$


Mathematics

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