Evaluate: $\int \frac{2 x}{\sqrt{(x^{2} + 1) (x^{2} + 3)}} dx$

Solution

$I = \int \frac{2 x}{(x^{2} + 1) (x^{2} + 3)} dx Let x^{2} = z ∴ 2 x dx = dz ∴ I = \int \frac{dz}{(z + 1) (z + 3)}$

By partial fraction,

$\frac{1}{(z + 1) (z + 3)} = \frac{A}{(z + 1)} + \frac{B}{(z + 3)} \Rightarrow 1 = A (z + 3) + B (z + 1) Putting z = - 3, we obtain : 1 = - 2 B B = - \frac{1}{2} ∴ A = \frac{1}{2}$ $∴ \frac{1}{(z + 1) (z + 3)} = \frac{\frac{1}{2}}{z + 1} + \frac{(- \frac{1}{2})}{z + 3} \Rightarrow \int \frac{dz}{(z + 1) (z + 3)} = \frac{1}{2} \int \frac{dz}{z + 1} - \frac{1}{2} \int \frac{dz}{z + 3} = \frac{1}{2} \log |z + 1| - \frac{1}{2} \log |z + 3| + C ∴ \int \frac{2 x dx}{(x^{2} + 1) (x^{2} + 3)} = \frac{1}{2} \log |x^{2} + 1| - \frac{1}{2} \log |x^{2} + 3| + C$

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Evaluate: $\int \frac{2 x}{\sqrt{(x^{2} + 1) (x^{2} + 3)}} dx$

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Evaluate: $\int \frac{2 x}{\sqrt{(x^{2} + 1) (x^{2} + 3)}} dx$