`int sqrt(9x^(2)+16)dx=?`

Options

A. `(x)/(2)sqrt(9x^(2)+16)+(8)/(3)log|3x+sqrt(9x^(2)+16)|+C`
B. `(x)/(2)sqrt(9x^(2)+16)-(8)/(3)log|3x+sqrt(9x^(2)+16)|+C`
C. `xsqrt(9x^(2)+16)+24log|3x+sqrt(9x^(2)+16)|+C`
D. None of these

Answer

A

Explanation

`int sqrt(x^(2)+a^(2))dx=(x)/(2)sqrt(x^(2)+a^(2))+(a^(2))/(2)log |x+sqrt(x^(2)+a^(2))|+C`
`therefore I=3int sqrt(x^(2)+(16)/(9))dx=3 int sqrt(x^(2)+((4)/(3))^(2))dx`
`=3{(x)/(2)sqrt(x^(2)+(16)/(9))+(8)/(9) log |x+sqrt(x^(2)+(16)/(9)|}+C`
`=(x)/(2) sqrt(x^(2)+16)+(8)/(3) log |3x+sqrt(9x^(2)+16)|+C.`