Advertisements
Advertisements
प्रश्न
उत्तर
We know that
\[\left| x + 1 \right| = \begin{cases}x + 1, & \text{if }x + 1 \geq 0 \\ - \left( x + 1 \right), & \text{if }x + 1 < 0\end{cases} = \begin{cases}x + 1, & \text{if }x \geq - 1 \\ - \left( x + 1 \right), & \text{if }x < - 1\end{cases}\]
\[\left| x - 1 \right| = \begin{cases}x - 1, & \text{if }x - 1 \geq 0 \\ - \left( x - 1 \right), & \text{if }x - 1 < 0\end{cases} = \begin{cases}x - 1, & \text{if }x \geq 1 \\ - \left( x - 1 \right), & \text{if }x < 1\end{cases}\]
When
When
When
\[\therefore \int_{- 1}^2 \left( \left| x + 1 \right| + \left| x \right| + \left| x - 1 \right| \right)dx\]
\[ = \int_{- 1}^0 \left( 2 - x \right)dx + \int_0^1 \left( x + 2 \right)dx + \int_1^2 3xdx\]
\[ = \left.\frac{\left( 2 - x \right)^2}{2 \times \left( - 1 \right)}\right|_{- 1}^0 + \left.\frac{\left( x + 2 \right)^2}{2}\right|_0^1 + \left.3 \times \frac{x^2}{2}\right|_1^2 \]
\[ = - \frac{1}{2}\left( 4 - 9 \right) + \frac{1}{2}\left( 9 - 4 \right) + \frac{3}{2}\left( 4 - 1 \right)\]
\[ = \frac{5}{2} + \frac{5}{2} + \frac{9}{2}\]
\[ = \frac{19}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following integral:
Evaluate the following integral:
Evaluate each of the following integral:
\[\int_a^b \frac{x^\frac{1}{n}}{x^\frac{1}{n} + \left( a + b - x \right)^\frac{1}{n}}dx, n \in N, n \geq 2\]
Prove that:
Evaluate each of the following integral:
Evaluate each of the following integral:
Evaluate each of the following integral:
The value of the integral \[\int\limits_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx\] is
If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals
\[\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx\]
\[\int\limits_1^3 \left( x^2 + 3x \right) dx\]
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
Evaluate the following:
Γ(4)
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
