Advertisements
Advertisements
प्रश्न
\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]
Advertisements
उत्तर
\[\int_{- a}^a \frac{x e^{x^2}}{1 + x^2} d x\]
\[\text{Let }f(x) = \frac{x e^{x^2}}{1 + x^2}\]
\[\text{Consider }f(-x) = - \frac{x e^{x^2}}{1 + x^2} = - f\left( x \right)\]
Thus f(x) is an odd function
Therefore,
\[ \int_{- a}^a \frac{x e^{x^2}}{1 + x^2} d x = 0\]
APPEARS IN
संबंधित प्रश्न
\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]
\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]
If f(x) is a continuous function defined on [−a, a], then prove that
Solve each of the following integral:
If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.
Evaluate :
\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\] equals
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
Evaluate : \[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\] .
\[\int\limits_1^2 x\sqrt{3x - 2} dx\]
\[\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx\]
\[\int\limits_0^1 \left| \sin 2\pi x \right| dx\]
\[\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx\]
\[\int\limits_0^4 x dx\]
Using second fundamental theorem, evaluate the following:
`int_0^(pi/2) sqrt(1 + cos x) "d"x`
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`
Find `int x^2/(x^4 + 3x^2 + 2) "d"x`
Find `int sqrt(10 - 4x + 4x^2) "d"x`
Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`
Verify the following:
`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`
