Solve $\int \frac{e^{x}(2-x^2)}{(1-x)\sqrt{1-x^2}}\mathrm dx$

Find $$\int \frac{e^{x}(2-x^2)}{(1-x)\sqrt{1-x^2}}\mathrm dx$$

Looking at the numerator, combined with the surd, you can get $$\int\frac{e^x(1+x)\left(\frac{1}{\sqrt{1-x^2}}+\sqrt{1-x^2}\right)}{1-x^2}\mathrm dx$$Then this begins to look like the quotient rule, since the denominator is $$\sqrt{1-x^2}^2$$, and in the numerator, $$1/\sqrt{1-x^2}$$ is almost like the derivative of the square root. However, it isn't quite that - there are extra terms. $$\color{red}{e^x\left(\sqrt{1-x^2}+\frac{x}{\sqrt{1-x^2}}\right)}+\frac{e^x}{\sqrt{1-x^2}}+e^x x\sqrt{1-x^2}$$ The red terms are accounted for by quotient rule (giving an integral of $$\frac{e^x}{\sqrt{1-x^2}}$$). But then what do we do with the remaining terms?

Hint

Observe that the exponent of $$1-x$$ is $$-\dfrac32$$

So, let us find $$\dfrac{d\left(e^x\dfrac{(1+x)^n}{\sqrt{1-x}}\right)}{dx}$$

Compare with the given expression to find the value of $$n$$

• Wow, that was surprisingly simple... With $n=1/2$, I suppose this means my method with the quotient rule was bound to fail! – John Doe Apr 4 at 1:55

Hint:

$$\dfrac{1+1-x^2}{(1-x)^{3/2}(1+x)^{1/2}}=\dfrac1{...}+f(x)$$

where $$f(x)=\dfrac{\sqrt{1+x}}{\sqrt{1-x}},$$

$$f'(x)=?$$

Recall $$\dfrac{d(e^xf(x))}{dx}=?$$

$$x=\cos2t,dx=?$$

$$-I=\int\dfrac{e^{\cos2t}(1+\sin^22t)}{\sin^2t}=e^{\cos2t}\csc^2t-\dfrac{d(e^{\cos2t})}{dt}(-\cot t)$$

$$=\dfrac{d(e^{\cos2t}(-\cot t))}{dt}$$