Mathematics

$$\int { \left( x+2 \right) \sqrt { 3x+5 }  } dx$$


SOLUTION

We have,

$$\int{\left( x+2 \right)\sqrt{3x+5}}dx$$


Let

$$ 3x+5=t\Rightarrow x=\dfrac{t-5}{3} $$

$$ 3dx=dt $$

$$ dx=\dfrac{dt}{3} $$


So,

$$ \int{\left( \dfrac{t-5}{3}+2 \right)}\sqrt{t}dx $$

$$ \Rightarrow \int{\left( \dfrac{t-5+6}{3} \right)}\sqrt{t}dx $$

$$ \Rightarrow \int{\left( \dfrac{t+1}{3} \right)\sqrt{t}dt} $$

$$ \Rightarrow \dfrac{1}{3}\int{\left( t+1 \right)\sqrt{t}}dt $$

$$ \Rightarrow \dfrac{1}{3}\int{t\sqrt{t}}dt+\dfrac{1}{3}\int{\sqrt{t}}dt $$

$$ \Rightarrow \dfrac{1}{3}\int{{{t}^{\frac{3}{2}}}dt}+\dfrac{1}{3}\int{{{t}^{\frac{1}{2}}}dt} $$


On integrating and we get,

$$ \Rightarrow \dfrac{1}{3}\dfrac{{{t}^{\frac{3}{2}+1}}}{\dfrac{3}{2}+1}+\dfrac{1}{3}\dfrac{{{t}^{\frac{1}{2}+1}}}{\dfrac{1}{2}+1}+C $$

$$ \Rightarrow \dfrac{1}{3}\dfrac{{{t}^{\frac{5}{2}}}}{\dfrac{5}{2}}+\dfrac{1}{3}\dfrac{{{t}^{\dfrac{}{2}}}}{\dfrac{3}{2}}+C $$

$$ \Rightarrow \dfrac{2{{t}^{\frac{5}{2}}}}{15}+\dfrac{2{{t}^{\frac{3}{2}}}}{9}+C $$


now, put 

$$t=3x+5$$

Then,

$$\Rightarrow \dfrac{2{{\left( 3x+5 \right)}^{\frac{5}{2}}}}{15}+\dfrac{2{{\left( 3x+5 \right)}^{\frac{3}{2}}}}{9}$$


Hence, this is the answer.

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