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Prove That: - Mathematics and Statistics

Sum

Prove that: int "dx"/(sqrt("x"^2 +"a"^2)) = log  |"x" +sqrt("x"^2 +"a"^2) | + "c"

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Solution

Let I = int  1/sqrt("x"^2 + "a"^2) "dx"

Put x = a tan θ ⇒ tan θ = "x"/"a"

∴ dx = a sec2 θ dθ

∴ I = int  1/ sqrt("a"^2 "tan"^2 theta +"a"^2) "a"  "sec"^2  theta  "d" theta

= int  ("a"."sec"^2 theta)/("a" sqrt(1+"tan"^2 theta))     "d"theta

= int  ("sec"^2 theta)/("sec" theta) "d"theta

= int  "sec" theta . "d" theta

= "log"   |"sec" theta +"tan" theta| +"c"_1

= "log" |"x"/"a" + sqrt("sec"^2 theta)| + "c"_1

= "log" | "x"/"a" + sqrt 1+ "tan"^2 theta | + "c"_1

="log" |"x" /"a" +sqrt(1+"x"^2/"a"^2)| +"c"_1

= "log" |"x"/"a" + sqrt( "a"^2 + "x"^2)/"a"| + "c"_1

= "log" |"x" +sqrt("x"^2 +"a"^2)| - "log" "a" + "c"_1

therefore int  1/sqrt("x"^2 + "a"^2) "dx" = "log" |"x" +sqrt("x"^2 +"a"^2)| - "log" "a" + "c" ,

where c = - log a +c1

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