समाकलन ज्ञात करें `int sqrt(x^2 - a^2) dx `

Options

A. `(xsqrt(x^2-a^2))/2-a^2/2log|x+sqrt(x^2-a^2)|+C`
B. `(sqrt(x^2-a^2))/2-a^2/2log|x+sqrt(x^2-a^2)|+C`
C. `(xsqrt(x^2-a^2))/2-log|x+sqrt(x^2-a^2)|+C`
D. इनमे से कोई नहीं

Answer

A

Explanation

`x = a sec theta` रखे, तो `dx= a sec theta tan theta d theta`
अब, `I = int sqrt(a^2 sec^2 theta - a^2) a sec theta tan theta d theta = a^2 int sec theta tan^2 theta d theta`
`=a^2 int sec theta ( sec^2 theta-1) d theta = a^2 int (sec^3 theta - sec theta) d theta" ...(1)"`
`=a^2 int sec^3 theta d theta - a^2 int sec theta d theta`
`=a^2[(sec theta tan theta)/2+1/2log | sec theta + tan theta|-log |sec theta + tan theta|]+c`
`=a[1/2 sec theta tan theta -1/2 log|sec theta+tan theta |]+c`
`=a^2[1/2*x/a*sqrt(x^2-a^2)/a-1/2log|x/a+sqrt(x^2-a^2)/a|]+c`
`=(xsqrt(x^2-a^2))/2-a^2/alog|x+sqrt(x^2-a^2)|+k`