Solve the integral applying the substitution $u^2=\frac{3x^2}{2}$. Then, take the square root of both sides, simplifying we have
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Isolate $dx$ in the previous equation
After replacing everything and simplifying, the integral results in
Simplify the expression
Apply the formula: $\int\frac{1}{1-x^2}dx$$=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)+C$, where $x=u$
Multiplying fractions $\frac{\frac{\sqrt{2}}{\sqrt{3}}}{2} \times \frac{1}{2}$
Replace $u$ with the value that we assigned to it in the beginning: $\frac{\sqrt{3}x}{\sqrt{2}}$
Multiplying the fraction by $\ln\left|\frac{\frac{\sqrt{3}x}{\sqrt{2}}+1}{\frac{\sqrt{3}x}{\sqrt{2}}-1}\right|$
Simplify the expression
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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