I'm doing some problems that involve identifying the best method of integration and then using that method. I'm given the following: $$\int\frac{dx}{(1-x^{2})^{\frac{3}{2}}} dx$$
I solved it using u-substitution: $$\int\frac{dx}{(1-x^{2})^{\frac{3}{2}}} dx = \int\frac{1}{(1-x^{2})^{\frac{3}{2}}} dx$$ $$u=1-x^2, du = -2x dx$$ $$-\frac{1}{2}\int\frac{1}{u^{\frac{3}{2}}}du = -\frac{1}{2}\int u^{-\frac{3}{2}}du$$ $$-\frac{1}{2}[-\frac{1}{2}u^{-\frac{1}{2}}] = \frac{1}{4}u^{-\frac{1}{2}} = \frac{1}{4\sqrt{u}}$$ $$= \frac{1}{4\sqrt{(1-x^{2})}} + C$$
My book, however, uses trigonometric substitution to solve this integral and gets an answer of: $$\frac{x}{1-x^{2}}+C$$
Did I do something wrong? Where did I mess up?