Advertisements
Advertisements
प्रश्न
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
उत्तर
\[\text{Here }a = 1, b = 4, f\left( x \right) = x^2 + x, h = \frac{4 - 1}{n} = \frac{3}{n}\]
Therefore,
\[ \int_1^4 \left( x^2 + x \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) + . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ f\left( 1 \right) + f\left( 1 + h \right) + . . . . . . . . . . + f\left( 1 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 1 + 1 + \left( 1 + h \right)^2 + \left( 1 + h \right) + \left( 1 + 2h \right)^2 + \left( 1 + 2h \right) + . . . . . . . . . + \left( 1 + \left( n - 1 \right)h \right)^2 + \left( 1 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 2n + h^2 \left( 1^2 + 2^2 + . . . . . . . . . . . . . . \left( n - 1 \right)^2 \right) + 2h\left( 1 + 2 + . . . . . . + \left( n - 1 \right) \right) + h\left( 1 + 2 + . . . . . . + \left( n - 1 \right) \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 2n + h^2 \frac{n\left( n - 1 \right)\left( 2n - 1 \right)}{6} + 3h\frac{n\left( n - 1 \right)}{2} \right]\]
\[ = \lim_{n \to 0 } \left[ 6 + \frac{9}{2}\left( 1 - \frac{1}{n} \right)\left( 2 - \frac{1}{n} \right) + \frac{9}{2}\left( 1 - \frac{1}{n} \right) \right]\]
\[ = 6 + 9 + \frac{9}{2} = \frac{27}{2}\]
APPEARS IN
संबंधित प्रश्न
If `f` is an integrable function such that f(2a − x) = f(x), then prove that
If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.
The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is
The value of the integral \[\int\limits_0^{\pi/2} \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx\] is
\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\] equals
If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
Evaluate : \[\int e^{2x} \cdot \sin \left( 3x + 1 \right) dx\] .
\[\int\limits_0^1 \tan^{- 1} x dx\]
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]
\[\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
Evaluate the following:
Γ(4)
If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
`int x^9/(4x^2 + 1)^6 "d"x` is equal to ______.
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
