English

CBSECommerce (English Medium) Class 12

Question Papers

Question Papers2890

Textbook Solutions20276

MCQ Online Mock Tests42

Important Solutions23871

Concept Notes & Videos601

Π / 2 ∫ − π / 2 Log ( 2 − Sin X 2 + Sin X ) D X

Advertisements

Advertisements

Question

\[\int\limits_{- \pi/2}^{\pi/2} \log\left( \frac{2 - \sin x}{2 + \sin x} \right) dx\]

Advertisements

Solution

\[Let\ I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \log\left( \frac{2 - \sin x}{2 + \sin x} \right) d x\]
\[Here, f\left( x \right) = log\left( \frac{2 - \sin x}{2 + \sin x} \right)\]
\[f\left( - x \right) = log\left( \frac{2 - \sin\left( - x \right)}{2 + \sin\left( - x \right)} \right) = log\left( \frac{2 + \sin x}{2 - \sin x} \right) = - log\left( \frac{2 - \sin x}{2 + \sin x} \right) = - f\left( x \right)\]
\[\text{Hence} f\left( x \right) \text{is an odd function}\]
\[ \therefore I = 0\]

shaalaa.com

Definite Integrals

Is there an error in this question or solution?

Chapter 19: Definite Integrals - Exercise 20.5 [Page 95]

APPEARS IN

R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12

Chapter 19 Definite Integrals
Exercise 20.5 | Q 28 | Page 95

RELATED QUESTIONS

\[\int\limits_0^1 \frac{x}{x + 1} dx\]

\[\int\limits_0^{\pi/6} \cos x \cos 2x\ dx\]

\[\int\limits_0^{\pi/2} \sin x \sin 2x\ dx\]

Evaluate the following definite integrals:

\[\int_0^\frac{\pi}{2} x^2 \sin\ x\ dx\]

\[\int\limits_0^{\pi/2} x \cos\ x\ dx\]

\[\int\limits_1^4 \frac{x^2 + x}{\sqrt{2x + 1}} dx\]

\[\int\limits_0^{2\pi} e^{x/2} \sin\left( \frac{x}{2} + \frac{\pi}{4} \right) dx\]

\[\int\limits_1^2 \frac{3x}{9 x^2 - 1} dx\]

\[\int\limits_0^1 \frac{e^x}{1 + e^{2x}} dx\]

\[\int\limits_0^1 x e^{x^2} dx\]

\[\int\limits_0^1 \tan^{- 1} x\ dx\]

\[\int\limits_{- 1}^1 5 x^4 \sqrt{x^5 + 1} dx\]

\[\int_0^\frac{\pi}{2} \frac{\cos^2 x}{1 + 3 \sin^2 x}dx\]

\[\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx\]

\[\int\limits_4^9 \frac{\sqrt{x}}{\left( 30 - x^{3/2} \right)^2} dx\]

\[\int\limits_0^a x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} dx\]

\[\int\limits_{- a}^a \sqrt{\frac{a - x}{a + x}} dx\]

\[\int\limits_0^\pi \frac{x \tan x}{\sec x \ cosec x} dx\]

\[\int\limits_3^5 \left( 2 - x \right) dx\]

\[\int\limits_a^b x\ dx\]

\[\int\limits_0^5 \left( x + 1 \right) dx\]

\[\int\limits_{- \pi/2}^{\pi/2} x \cos^2 x\ dx .\]

\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.

\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals

\[\int\limits_{\pi/6}^{\pi/3} \frac{1}{\sin 2x} dx\] is equal to

\[\lim_{n \to \infty} \left\{ \frac{1}{2n + 1} + \frac{1}{2n + 2} + . . . + \frac{1}{2n + n} \right\}\] is equal to

Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .

\[\int\limits_0^4 x\sqrt{4 - x} dx\]

\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]

\[\int\limits_1^3 \left| x^2 - 4 \right| dx\]

\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]

\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]

\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]

Evaluate the following:

`int_(-1)^1 "f"(x) "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x < 0):}`

Evaluate the following using properties of definite integral:

`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`

Evaluate the following:

`Γ (9/2)`

Choose the correct alternative:

If n > 0, then Γ(n) is

Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`

Find: `int logx/(1 + log x)^2 dx`

Notifications