Advertisements
Advertisements
प्रश्न
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
Advertisements
उत्तर
\[\int_0^\frac{\pi}{3} \frac{\cos x}{3 + 4\sin x} d x\]
\[Let, \sin x = t \Rightarrow \cos x dx = dt\]
\[\text{When, }\sin x \to 0 ; t \to 0\]
\[\text{And }\sin x \to \frac{\pi}{3} ; t \to \frac{\sqrt{3}}{2}\]
\[ = \int_0^\frac{\sqrt{3}}{2} \frac{dt}{3 + 4t}\]
\[ = \frac{1}{4}\log \left[ 3 + 4t \right]_0^\frac{\sqrt{3}}{2} \]
\[ = \frac{1}{4}\log\left[ \log\left( 3 + 2\sqrt{3} \right) - \log\left( 3 + 0 \right) \right]\]
\[ = \frac{1}{4}\log\left[ \log\left( 2\sqrt{3} + 3 \right) - \log\left( 3 \right) \right]\]
\[ = \frac{1}{4}\left( \log\frac{2\sqrt{3} + 3}{3} \right)\]
APPEARS IN
संबंधित प्रश्न
Evaluate each of the following integral:
Evaluate the following integral:
If f(2a − x) = −f(x), prove that
If f is an integrable function, show that
\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]
If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.
The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
\[\int\limits_0^1 \cos^{- 1} x dx\]
\[\int\limits_0^{1/\sqrt{3}} \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]
\[\int\limits_0^{\pi/2} x^2 \cos 2x dx\]
\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
\[\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx\]
Using second fundamental theorem, evaluate the following:
`int_1^2 (x - 1)/x^2 "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
Choose the correct alternative:
Γ(n) is
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
What is the meaning of an indefinite integral?
