Advertisements
Advertisements
प्रश्न
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that \[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]
Advertisements
उत्तर
\[\int_a^b xf\left( x \right)dx\]
\[ = \int_a^b \left( a + b - x \right)f\left( a + b - x \right)dx ..................\left[ \int_a^b f\left( x \right)dx = \int_a^b f\left( a + b - x \right)dx \right]\]
\[ = \int_a^b \left( a + b - x \right)f\left( x \right)dx ..................\left[ f\left( a + b - x \right) = f\left( x \right) \right]\]
\[ \therefore \int_a^b xf\left( x \right)dx = \int_a^b \left( a + b \right)f\left( x \right)dx - \int_a^b xf\left( x \right)dx\]
\[\Rightarrow \int_a^b xf\left( x \right)dx + \int_a^b xf\left( x \right)dx = \left( a + b \right) \int_a^b f\left( x \right)dx\]
\[ \Rightarrow 2 \int_a^b xf\left( x \right)dx = \left( a + b \right) \int_a^b f\left( x \right)dx\]
\[ \Rightarrow \int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]
APPEARS IN
संबंधित प्रश्न
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
If f is an integrable function, show that
If f(x) is a continuous function defined on [−a, a], then prove that
Evaluate each of the following integral:
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
\[\int\limits_0^1 x \left( \tan^{- 1} x \right)^2 dx\]
\[\int\limits_0^{\pi/4} e^x \sin x dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx\]
\[\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx\]
\[\int\limits_{- 1}^1 e^{2x} dx\]
Find : `∫_a^b logx/x` dx
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Evaluate the following:
`int_0^2 "f"(x) "d"x` where f(x) = `{{:(3 - 2x - x^2",", x ≤ 1),(x^2 + 2x - 3",", 1 < x ≤ 2):}`
Evaluate the following using properties of definite integral:
`int_(-1)^1 log ((2 - x)/(2 + x)) "d"x`
Evaluate the following:
`Γ (9/2)`
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
Choose the correct alternative:
`int_0^1 (2x + 1) "d"x` is
Choose the correct alternative:
`int_0^oo "e"^(-2x) "d"x` is
If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.
Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`
What is an antiderivative of \[x\]?
