Advertisements
Advertisements
Question
Prove that:
`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`
Advertisements
Solution
Let I = `int sqrt(x^2 + a^2)dx`
= `int sqrt(x^2 + a^2)*1dx`
= `sqrt(x^2 + a^2) int 1dx - int[d/dx(sqrt(x^2 + a^2))*int1dx]dx`
= `sqrt(x^2 + a^2)*x - int (2x)/(2sqrt(x^2 + a^2))*x dx`
= `x*sqrt(x^2 + a^2) - int ((x^2 + a^2) - a^2)/sqrt(x^2 + a^2)dx`
= `x*sqrt(x^2 + a^2) - int ((x^2 + a^2)/sqrt(x^2 + a^2) - a^2/sqrt(x^2 + a^2))dx`
= `x*sqrt(x^2 + a^2) - int sqrt(x^2 + a^2)dx + a^2 int 1/sqrt(x^2 + a^2)dx`
∴ I = `x*sqrt(x^2 + a^2) - I + a^2log|x + sqrt(x^2 + a^2)| + c_1`
∴ 2I = `x*sqrt(x^2 + a^2) + a^2 log|x + sqrt(x^2 + a^2)| + c_1`
∴ I = `x/2 sqrt(x^2 + a^2) + a^2/2 log|x + sqrt(x^2 + a^2)| + c_1/2`
∴ `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, "where" c = c_1/2`
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in x (log x)2.
Integrate the function in (x2 + 1) log x.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `e^x (1/x - 1/x^2)`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
∫ x log x dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Evaluate `int 1/(x log x) "d"x`
∫ log x · (log x + 2) dx = ?
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`inte^x sinx dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Prove that `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`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Integration by parts is a method of integration based on which rule of differentiation?
