{1}{{x+a}+{x+b}} - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

..... Miscellaneous Exercise .....
Integrate the functions in Exercises 229 to 252.

Question : 230 of 271

Marks: +1, -0

\frac{1}{\sqrt{x+a}+\sqrt{x+b}}

Solution:

Let

I = \int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} \, dx

= \int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} \times \frac{\sqrt{x+a}-\sqrt{x+b}}{\sqrt{x+a}-\sqrt{x+b}} \, dx

= \int \frac{\sqrt{x+a}-\sqrt{x+b}}{(\sqrt{x+a})^2 - (\sqrt{x+b})^2} \, dx

= \frac{1}{a-b} \int \left[ (x+a)^{1/2} - (x+b)^{1/2} \right] dx

= \frac{1}{a-b} \left[ \frac{(x+a)^{3/2}}{\frac{3}{2} \cdot 1} - \frac{(x+b)^{3/2}}{\frac{3}{2} \cdot 1} \right] + C

= \frac{2}{3(a-b)} \left[ (x+a)^{3/2} - (x+b)^{3/2} \right] + C

Go to Question: