Advertisements
Advertisements
प्रश्न
Evaluate each of the following integral:
Advertisements
उत्तर
\[I = \int_0^1 x e^{x^2} dx\]
\[ = \frac{1}{2} \int_0^1 e^{x^2} 2xdx\]
Put \[x^2 = z\]
When \[x \to 0, z \to 0\]
When \[x \to 1, z \to 1\]
\[\therefore I = \frac{1}{2} \int_0^1 e^z dz\]
\[ = \frac{1}{2} \left.\times {e^z}\right|_0^1 \]
\[ = \frac{1}{2}\left( e - e^0 \right)\]
\[ = \frac{1}{2}\left( e - 1 \right)\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals:
Evaluate the following integral:
If f is an integrable function, show that
Evaluate each of the following integral:
Evaluate each of the following integral:
\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.
The value of \[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is __________ .
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
\[\int\limits_0^4 x\sqrt{4 - x} dx\]
\[\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]
\[\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx\]
\[\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx\]
\[\int\limits_0^\pi \cos 2x \log \sin x dx\]
\[\int\limits_0^1 \cot^{- 1} \left( 1 - x + x^2 \right) dx\]
Choose the correct alternative:
If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is
Choose the correct alternative:
`Γ(3/2)`
Which integral has Constant C present?
What is an antiderivative of \[2x+1\]?
