Advertisements
Advertisements
Question
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Advertisements
Solution
\[\int_a^b xf\left( x \right)dx = \int_a^b \left( a + b - x \right)f\left( a + b - x \right)dx ................\left[ \int_a^b f\left( x \right)dx = \int_a^b f\left( a + b - x \right)dx \right]\]
\[ \Rightarrow \int_a^b xf\left( x \right)dx = \int_a^b \left( a + b - x \right)f\left( x \right)dx .....................\left[ f\left( a + b - x \right) = f\left( x \right) \right]\]
\[ \Rightarrow \int_a^b xf\left( x \right)dx = \int_a^b \left( a + b \right)f\left( x \right)dx - \int_a^b xf\left( x \right)dx\]
\[ \Rightarrow 2 \int_a^b xf\left( x \right)dx = \left( a + b \right) \int_a^b f\left( x \right)dx\]
\[ \Rightarrow \int_a^b xf\left( x \right)dx = \left( \frac{a + b}{2} \right) \int_a^b f\left( x \right)dx\]
APPEARS IN
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
`int_0^2 e^x dx` = ______.
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_2^3 x/(x^2 - 1)` dx = ______
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_0^pi x sin^2x dx` = ______
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int_0^1 1/(2x + 5) dx` = ______.
If `int_a^b x^3 dx` = 0, then `(x^4/square)_a^b` = 0
⇒ `1/4 (square - square)` = 0
⇒ b4 – `square` = 0
⇒ (b2 – a2)(`square` + `square`) = 0
⇒ b2 – `square` = 0 as a2 + b2 ≠ 0
⇒ b = ± `square`
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_4^9 1/sqrt(x)dx` = ______.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
`int_(pi"/"11)^(9pi"/"22) (dx)/(1 + sqrttan x)` =
