Advertisements
Advertisements
प्रश्न
उत्तर
\[\text{We have}, \]
\[I = \int\limits_0^2 x\left[ x \right] dx\]
\[\text{We know that}, \]
\[x\left[ x \right] = \begin{cases}x \times 0&,& 0 < x < 1\\x \times 1&,& 1 < x < 2\end{cases}\]
\[i . e . , \]
\[x\left[ x \right] = \begin{cases}0&,& 0 < x < 1\\x&,& 1 < x < 2\end{cases}\]
\[ \therefore I = \int\limits_0^2 x\left[ x \right] dx\]
\[ = \int\limits_0^1 x\left[ x \right] dx + \int\limits_1^2 x\left[ x \right] dx\]
\[ = \int\limits_0^1 \left( 0 \right) dx + \int\limits_1^2 \left( x \right) dx\]
\[ = 0 + \left[ \frac{x^2}{2} \right]_1^2 \]
\[ = \frac{2^2}{2} - \frac{1^2}{2}\]
\[ = \frac{4}{2} - \frac{1}{2}\]
\[ = \frac{3}{2}\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
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 \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is
The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Choose the correct alternative:
`int_0^1 (2x + 1) "d"x` is
Choose the correct alternative:
If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is
Choose the correct alternative:
Γ(1) is
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.
`int "e"^x ((1 - x)/(1 + x^2))^2 "d"x` is equal to ______.
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
