Advertisements
Advertisements
प्रश्न
Evaluate each of the following integral:
Advertisements
उत्तर
\[\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx\]
\[ = - \int_0^\frac{\pi}{2} e^x \left[ \cos x + \left( - \sin x \right) \right]dx\]
\[ = \left.- {e^x \cos x}\right|_0^\frac{\pi}{2} .............\left\{ \int e^x \left[ f\left( x \right) + f'\left( x \right) \right]dx = e^x f\left( x \right) + C \right\}\]
\[ = - \left( e^\frac{\pi}{2} \cos\frac{\pi}{2} - e^0 \cos0 \right)\]
\[ = - \left( e^\frac{\pi}{2} \times 0 - 1 \times 1 \right)\]
\[ = - \left( 0 - 1 \right)\]
\[ = 1\]
संबंधित प्रश्न
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
Evaluate the following integral:
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 f (x) is a continuous function defined on [0, 2a]. Then, prove that
Evaluate :
The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .
`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx\]
\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx\]
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
Choose the correct alternative:
Γ(n) is
Choose the correct alternative:
Γ(1) is
Choose the correct alternative:
`int_0^oo x^4"e"^-x "d"x` is
`int "e"^x ((1 - x)/(1 + x^2))^2 "d"x` is equal to ______.
