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 each of the following integral:
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
\[\int\limits_1^2 x\sqrt{3x - 2} dx\]
\[\int\limits_0^{\pi/2} x^2 \cos 2x dx\]
Evaluate the following integrals :-
\[\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx\]
\[\int\limits_0^{\pi/4} \tan^4 x dx\]
Using second fundamental theorem, evaluate the following:
`int_0^(1/4) sqrt(1 - 4) "d"x`
Evaluate the following:
Γ(4)
Evaluate the following:
`int_0^oo "e"^(-4x) x^4 "d"x`
Evaluate the following:
`int_0^oo "e"^(- x/2) x^5 "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
Choose the correct alternative:
Γ(1) is
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`
What is an antiderivative of \[2x+1\]?
