Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\left| x \right| = \begin{cases} - x &,& - 1 < x < 0\\ x &,& 0 < x < 1\end{cases}\]
\[ \therefore x\left| x \right| = \begin{cases} - x^2 &,& - 1 < x < 0\\ x^2 &,& 0 < x < 1\end{cases}\]
\[Now\, \int_{- 1}^1 x\left| x \right| d x\]
\[ = \int_{- 1}^0 - x^2 dx + \int_0^1 x^2 dx\]
\[ = - \int_{- 1}^0 x^2 dx + \int_0^1 x^2 dx\]
\[ = - \left[ \frac{x^3}{3} \right]_{- 1}^0 + \left[ \frac{x^3}{3} \right]_0^1 \]
\[ = - \left( 0 + \frac{1}{3} \right) + \left( \frac{1}{3} - 0 \right)\]
\[ = 0 - \frac{1}{3} + \frac{1}{3} - 0\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals:
Evaluate the following integral:
The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is
The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is
The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
\[\int\limits_0^1 \cos^{- 1} x dx\]
\[\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx\]
\[\int\limits_0^{\pi/2} x^2 \cos 2x dx\]
\[\int\limits_0^1 \left| \sin 2\pi x \right| dx\]
\[\int\limits_0^{2\pi} \cos^7 x dx\]
\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
\[\int\limits_2^3 e^{- x} dx\]
Evaluate the following using properties of definite integral:
`int_0^1 log (1/x - 1) "d"x`
Evaluate the following using properties of definite integral:
`int_0^1 x/((1 - x)^(3/4)) "d"x`
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
`int x^3/(x + 1)` is equal to ______.
