Question

√x²-a² dx =x÷2 √x²-a²+a²log |x+√x²-a²| + c

Answer 1

To find---> ∫ √( x² - a² ) dx

Solution--->

1) plzzz refer to the attachment

2) We have a formula of intregation by parts as follows

∫ (1) ( 2 ) dx = (1) ∫ (2) dx - ∫ { d/dx (1) ∫2 dx } dx

3) We choose √(x² - a² ) as 1st function and 1 as second function

4) Then we apply two formulee

a) ∫ 1 dx = x

b) d/dx ( √x ) = 1 / 2√x

5) In 7th step we add and subtract a² and then in next step seperate in to two fractions.

6) In 11th step we put √(x² - a²) = I

7) In 12th step we apply a formula of intregation

∫ 1 / √(x² - a²) dx = log ( x + √(x² - a²)

8) then we get answer