Advertisements
Advertisements
प्रश्न
उत्तर
\[Let\ I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{1}{1 + \sin x} d x . Then, \]
\[I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{1}{1 + \sin x} \times \frac{1 - \sin x}{1 - \sin x} d x\]
\[ \Rightarrow I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{1 - \sin x}{1 - \sin^2 x} dx\]
\[ \Rightarrow I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{1 - \sin x}{\cos^2 x} dx\]
\[ \Rightarrow I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \left( \frac{1}{\cos^2 x} - \frac{\sin x}{\cos^2 x} \right) dx\]
\[ \Rightarrow I = \int_{- \frac{\pi}{4}}^\frac{\pi}{4} \left( \sec^2 x - \sec x \tan x \right) dx\]
\[ \Rightarrow I = \left[ \tan x - \sec x \right]_{- \frac{\pi}{4}}^\frac{\pi}{4} \]
\[ \Rightarrow I = \left( 1 - \sqrt{2} \right) - \left( - 1 - \sqrt{2} \right)\]
\[ \Rightarrow I = 2\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals:
Evaluate each of the following integral:
\[\int_a^b \frac{x^\frac{1}{n}}{x^\frac{1}{n} + \left( a + b - x \right)^\frac{1}{n}}dx, n \in N, n \geq 2\]
If \[\int_0^a \frac{1}{4 + x^2}dx = \frac{\pi}{8}\] , find the value of a.
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is ________ .
The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is
\[\int\limits_0^{2a} f\left( x \right) dx\] is equal to
The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is
\[\int\limits_1^2 x\sqrt{3x - 2} dx\]
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^{\pi/4} e^x \sin x dx\]
Evaluate the following:
Γ(4)
Choose the correct alternative:
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
Find: `int logx/(1 + log x)^2 dx`
