Advertisements
Advertisements
प्रश्न
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Advertisements
उत्तर
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
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Write a value of
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate `int(3x^2 - 5)^2 "d"x`
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
`int (f^'(x))/(f(x))dx` = ______ + c.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
