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Q23  Integrate the functions \frac{5x + 3 }{\sqrt { x^2 + 4x +10 }}

Answers (1)

best_answer

let 
5x+3 = A\frac{d}{dx}(x^2+4x+10)+B = A(2x+4)+B
On comparing, we get

A =5/2 and B = -7

\int \frac{5x+3}{\sqrt{x^2+4x+10}}dx = \frac{5}{2}\int \frac{2x+4}{\sqrt{x^2+4x+10}}dx-7\int \frac{dx}{\sqrt{x^2+4x+10}}dxI = 5/2I_1-7I_2...........................................(i)

\\\Rightarrow I_1\\ \int \frac{2x+4}{\sqrt{x^2+4x+10}}dx
put 
x^2+4x+10= t \Rightarrow (2x+4)dx = dt

=\int \frac{dt}{\sqrt{t}}=2\sqrt{t}=2\sqrt{x^2+4x+10}

\\\Rightarrow I_2\\ =\int \frac{1}{\sqrt{x^2+4x+10}}dx \\ =\int \frac{1}{\sqrt{(x+2)^2+(\sqrt{6})^2}}dx\\ =\log \left | (x+2)+\sqrt{x^2+4x+10} \right |

I = 5\sqrt{x^2+4x+10}-7\log\left | (x+2)+\sqrt{x^2+4x+10} \right |+C

Posted by

manish

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