Advertisements
Advertisements
प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Advertisements
उत्तर
`Let I =int1/(x^2sqrt(a^2+x^2)dx`
`Put x = a tantheta`
Differentiating w.r.t. theta we get
`dx = a sec^2 theta d theta`
`theta=tan^-1(x/a)`
`I=int(asec^2theta d theta)/(a^2tan^2thetasqrt(a^2+a^2tan^2theta))`
`=1/a^2intsectheta/tan^2theta d theta`
`=1/a^2intcostheta/sin^2thetad theta`
`=1/a^2intcosecthetacotthetad theta`
`I=-1/a^2cosectheta+c ....(i)`
`But tantheta=x/a`
`cottheta`=a/x`
`cosec^2theta`=1+cot^2theta`
`cosec^2theta=1+a^2/x^2`
`cosec^2theta=(x^2+a^2)/x^2`
`cosectheta=sqrt(x^2_a^2)/x.........(ii)`
`I=-1/a^2sqrt(x^2+a^2)/x+c `
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Integrate the functions:
`sin x/(1+ cos x)`
Solve:
dy/dx = cos(x + y)
Write a value of\[\int \log_e x\ dx\].
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).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` =
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate `int 1/((2"x" + 3))` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int 1/(sqrt("x") + "x")` dx
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int dx/(1 + e^-x)` = ______
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int cos^3x dx` = ______.
`int (logx)^2/x dx` = ______.
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int(cos 2x)/sinx dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
