Advertisements
Advertisements
Question
Evaluate the following integral:
Advertisements
Solution
\[\text{Let I} = \int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta\]
Consider
\[f\left( - \theta \right) = \log\left( \frac{a - \sin\left( - \theta \right)}{a + \sin\left( - \theta \right)} \right)\]
\[ = \log\left( \frac{a + \sin\theta}{a - \sin\theta} \right) ............\left[ \sin\left( - x \right) = - \sin x \right]\]
\[ = \log \left( \frac{a - \sin\theta}{a + \sin\theta} \right)^{- 1} \]
\[ = - \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right) ..............\left[ \log a^b = b\log a \right]\]
\[ = - f\left( \theta \right)\]
\[\therefore f\left( - \theta \right) = - f\left( \theta \right)\]
\[ \Rightarrow I = \int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta = 0 .................\left[ \int_{- a}^a f\left( x \right)dx = \begin{cases}2 \int_0^a f\left( x \right)dx, & \text{if }f\left( - x \right) = f\left( x \right) \\ 0, & \text{if }f\left( - x \right) = - f\left( x \right)\end{cases} \right]\]
APPEARS IN
RELATED QUESTIONS
If f(x) is a continuous function defined on [−a, a], then prove that
Evaluate each of the following integral:
Write the coefficient a, b, c of which the value of the integral
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
Evaluate : \[\int\frac{dx}{\sin^2 x \cos^2 x}\] .
\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]
\[\int\limits_0^{\pi/4} \tan^4 x dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]
\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]
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 integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
Choose the correct alternative:
If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
