Advertisements
Advertisements
प्रश्न
उत्तर
\[\int_a^b f\left( x \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) . . . . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]
\[\text{where }h = \frac{b - a}{n}\]
\[\text{Here }a = 1, b = 3, f\left( x \right) = 3x - 2, h = \frac{3 - 1}{n} = \frac{2}{n}\]
Therefore,
\[I = \int_1^3 \left( 3x - 2 \right) d x\]
\[ = \lim_{h \to 0} h\left[ f\left( 1 \right) + f\left( 1 + h \right) + . . . . . . . . . . . . . . . . . . . . + f\left( 1 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ \left( 3 - 2 \right) + \left( 3 + 3h - 2 \right) + \left( 3 + 6h - 2 \right) . . . . . . . . . . . . . . . + \left( 3\left( n - 1 \right)h + 3 - 2 \right) \right]\]
\[ = \lim_{h \to 0} h\left[ n + 3h\left( 1 + 2 + 3 . . . . . . . . . + \left( n - 1 \right) \right) \right]\]
\[ = \lim_{h \to 0} h\left[ n + 3h\frac{n\left( n - 1 \right)}{2} \right]\]
\[ = \lim_{n \to \infty} \frac{2}{n}\left[ n + 3n - 3 \right]\]
\[ = \lim_{n \to \infty} 2\left( 4 - \frac{3}{n} \right)\]
\[ = 8\]
APPEARS IN
संबंधित प्रश्न
If f is an integrable function, show that
If \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.
If \[\int_0^a \frac{1}{4 + x^2}dx = \frac{\pi}{8}\] , find the value of a.
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`
The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
\[\int\limits_0^{\pi/4} \sin 2x \sin 3x dx\]
\[\int\limits_0^1 \log\left( 1 + x \right) dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx\]
\[\int\limits_0^1 \left| 2x - 1 \right| dx\]
\[\int\limits_0^4 x dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
\[\int\limits_{- 1}^1 e^{2x} dx\]
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
`int x^3/(x + 1)` is equal to ______.
