Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[Let\, I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right) d \theta\]
\[Here\, f\left( \theta \right) = \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)\]
\[Consider\, f\left( - \theta \right) = \log\left[ \frac{a - \sin\left( - \theta \right)}{a + \sin\left( - \theta \right)} \right] = - \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right) = - f\left( \theta \right)\]
\[i . e . , f\left( \theta \right) \text{is odd function} . \]
\[\text{Therefore}, I = 0\]
APPEARS IN
संबंधित प्रश्न
If f (x) is a continuous function defined on [0, 2a]. Then, prove that
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
\[\int\limits_0^{2a} f\left( x \right) dx\] is equal to
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is
\[\int\limits_1^2 x\sqrt{3x - 2} dx\]
\[\int\limits_0^\pi \frac{x \sin x}{1 + \cos^2 x} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]
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_0^1 x/((1 - x)^(3/4)) "d"x`
Choose the correct alternative:
If n > 0, then Γ(n) is
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
Evaluate the following:
`int ((x^2 + 2))/(x + 1) "d"x`
`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.
`int "e"^x ((1 - x)/(1 + x^2))^2 "d"x` is equal to ______.
