Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[Let\ I = \int_a^b \frac{f\left( x \right)}{f\left( x \right) + f\left( a + b - x \right)} d x ................(1)\]
\[ = \int_a^b \frac{f\left( a + b - x \right)}{f\left( a + b - x \right) + f\left( a + b - a - b + x \right)} d x\]
\[ = \int_a^b \frac{f\left( a + b - x \right)}{f\left( a + b - x \right) + f\left( x \right)} d x\]
\[ \therefore I = \int_a^b \frac{f\left( a + b - x \right)}{f\left( x \right) + f\left( a + b - x \right)} d x ...............(2)\]
\[\text{Adding (1) and (2) we get}\]
\[2I = \int_a^b \left[ \frac{f\left( x \right)}{f\left( x \right) + f\left( a + b - x \right)} + \frac{f\left( a + b - x \right)}{f\left( x \right) + f\left( a + b - x \right)} \right] d x\]
\[ = \int_a^b \frac{f\left( x \right) + f\left( a + b - x \right)}{f\left( x \right) + f\left( a + b - x \right)} dx\]
\[ = \left[ x \right]_a^b \]
\[ = b - a\]
\[\text{Hence, }I = \frac{b - a}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate each of the following integral:
Evaluate each of the following integral:
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals
\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]
\[\int\limits_0^1 \left| 2x - 1 \right| dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Evaluate the following:
`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`
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 integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1
Find `int sqrt(10 - 4x + 4x^2) "d"x`
If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.
What is the meaning of an indefinite integral?
Evaluate \[\int_{0}^{1}x\,dx\].
