Advertisements
Advertisements
प्रश्न
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 each of the following integral:
Write the coefficient a, b, c of which the value of the integral
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
\[\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\] equals
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
Evaluate: \[\int\limits_{- \pi/2}^{\pi/2} \frac{\cos x}{1 + e^x}dx\] .
Evaluate : \[\int e^{2x} \cdot \sin \left( 3x + 1 \right) dx\] .
\[\int\limits_0^1 \log\left( 1 + x \right) dx\]
\[\int\limits_0^1 \left| 2x - 1 \right| dx\]
\[\int\limits_0^{2\pi} \cos^7 x dx\]
\[\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx\]
Evaluate the following:
`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`
Evaluate the following:
f(x) = `{{:("c"x",", 0 < x < 1),(0",", "otherwise"):}` Find 'c" if `int_0^1 "f"(x) "d"x` = 2
Evaluate the following:
`Γ (9/2)`
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`
