Advertisements
Advertisements
Question
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
Solution
`LHS=int_0^(2a)f(x)dx=int_0^af(x)dx+int_a^(2a)f(x)dx.........(1)`
Substitute x = a + t in the second integral
dx=dt
When x = a, t = 0.
When x = 2a, t = a.
`thereforeint_a^(2a)f(x)dx=int_0^af(a+t)dt`
`=int_0^af(a+(a-t))dt (therefore int_0^af(x)dx=int_0^a f(a-x)dx)`
`=int_0^af(2a-t)dt`
`int_a^(2a)f(x)dx=int_0^af(2a-x)dx (therefore int_0^af(t)dt=int_0^af(x)dx)`
Using the above in (1), we get
`int_0^(2a)f(x)dx=int_0^af(x)dx+int_a^(2a)f(x)dx`
`=int_0^af(x)dx+int_0^af(2a-x)dx=RHS ("Proved")`
APPEARS IN
RELATED QUESTIONS
Evaluate : `intsec^nxtanxdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_0^1 "e"^(2x) "d"x` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^1 (1 - x)^5`dx = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_0^{pi/2} cos^2x dx` = ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
