A Worthy Indefinite Integral

Evaluate \displaystyle\int \frac{1+\frac{\alpha}{2}x}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx, \; \alpha_1 \ne \alpha_2.


\displaystyle\int \frac{1+\frac{\alpha}{2}x}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx

= \displaystyle\int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx + \frac{\alpha}{2}\displaystyle\int\frac{x}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx

= \displaystyle\int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx + \frac{\alpha}{2}\cdot\left(\frac{1}{-2}\right)\displaystyle\int\frac{-2x+(\alpha_1+\alpha_2)-(\alpha_1+\alpha2)}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx

= \displaystyle\int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx - \frac{\alpha}{4}\left(\displaystyle\int\frac{-2x+(\alpha_1+\alpha_2)}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx - \displaystyle\int\frac{\alpha_1+\alpha2}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx\right)

=\boxed{ \displaystyle\int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx} - \frac{\alpha}{4}\displaystyle\int\frac{-2x+(\alpha_1+\alpha_2)}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx +\boxed{\frac{\alpha(\alpha_1+\alpha_2)}{4}\displaystyle\int\frac{1}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx}

= \boxed{(1+\frac{\alpha(\alpha_1+\alpha_2)}{4})\displaystyle\int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx}-\frac{\alpha}{4}\displaystyle\int\frac{-2x + (\alpha_1+\alpha2)}{\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}}\;dx

= \left(1+\frac{\alpha(\alpha_1+\alpha_2)}{4}\right)\displaystyle\int\frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx -\frac{\alpha}{4}\cdot\left(\frac{1}{1-\frac{1}{2}}\right)\sqrt{-x^2+(\alpha_1+\alpha_2)x-\alpha_1\alpha_2}

= \left(1+\frac{\alpha(\alpha_1+\alpha_2)}{4}\right)\displaystyle\int\frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx -\frac{\alpha}{2}\sqrt{-(x-\alpha_1)(x-\alpha_2)}

By \int \frac{1}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx=\arcsin\left(\frac{2x-(\alpha_1+\alpha_2)}{|\alpha_2-\alpha_1|}\right) (see “Integral: The CAS & I“),

= \left(1+\frac{\alpha(\alpha_1+\alpha_2)}{4}\right) \arcsin\left(\frac{2x-(\alpha_1+\alpha_2)}{|\alpha_2-\alpha_1|}\right)-\frac{\alpha}{2}\sqrt{-(x-\alpha_1)(x-\alpha_2)}


Exercise-1 Evaluate \displaystyle\int \frac{1+\frac{\alpha}{2}x}{\sqrt{-(x-\alpha_1)(x-\alpha_2)}}\;dx, \alpha_1 \ne \alpha_2 using a CAS.

Advertisement

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s