Mathematics

$$\int { \dfrac { \sqrt { 4+{ x }^{ 2 } }  }{ { x }^{ 6 } } dx } =\dfrac { { \left( a+{ x }^{ 2 } \right)  }^{ 3/2 }\left( { x }^{ 2 }-b \right)  }{ 120{ x }^{ 5 } }+C$$ then $$a+b$$ equals to:


SOLUTION
$$\displaystyle\int \dfrac{\sqrt{4+x^{2}}}{x^{6}}dx$$
 Put $$x=\dfrac{1}{t}\implies dx=-\dfrac{1}{t^{2}}dt$$
$$\displaystyle\int \dfrac{\sqrt{4+\dfrac{1}{t^{2}}}}{\dfrac{1}{t^{6}}}\times \dfrac{-dt}{t^{2}}=-\displaystyle \int t^{3}\sqrt{4t^{2}+1}dt$$
Put $$4{t^{2}}+1=l^{2}\implies t{dt}=\dfrac{l{dl}}{4}$$
$$\displaystyle\int -\dfrac{l^{2}-1}{4}\times \dfrac{l^{2}dl}{4}=-\dfrac{1}{16}\displaystyle\int (l^{4}-l^{2})dl=-\dfrac{l^{3}(3{l^{2}-5})}{240}=\dfrac{(4+x^{2})^{\dfrac{3}{2}}(x^{2}-\dfrac{3}{2})}{120{x^{5}}}$$
$$\implies a=4,b=\dfrac{3}{2}\implies a+b=4+\dfrac{3}{2}=\dfrac{11}{2}$$
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