Advertisements
Advertisements
प्रश्न
Evaluate the integral by using substitution.
`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`
Advertisements
उत्तर
Let `I = int_1^2 e^(2x) (1/x - 1/(2x^2)) dx`
Put 2x = t
⇒ 2dx = dt
When x = 1, t = 2
And when x = 2, t = 4
∴ `I = 1/2 int_2^4 e^t (2/t - (1 xx4)/(2t^2)) dt`
`= 1/2 int_2^4 e^t (2/t - 2/t^2) dt`
`= int_2^4 e^t* (1/t - 1/t^2) dt`
`= int_2^4 e^t *[1/t + d/dt (1/t)] dt`
`= [e^t * 1/t]_2^4 = 1/4 e^4 - e^2/2`
`= e^2/2 (e^2/2 - 1)`
or `(e^2 (e^2 - 2))/4`
APPEARS IN
संबंधित प्रश्न
Evaluate :`int_0^(pi/2)1/(1+cosx)dx`
Evaluate: `int1/(xlogxlog(logx))dx`
Evaluate `∫_0^(3/2)|x cosπx|dx`
Evaluate: `intsinsqrtx/sqrtxdx`
The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.
Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`
Evaluate of the following integral:
(i) \[\int x^4 dx\]
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate:
Evaluate:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate each of the following integral:
Evaluate each of the following integral:
Evaluate each of the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate: `int_ e^x ((2+sin2x))/cos^2 x dx`
Evaluate: `int_-π^π (1 - "x"^2) sin "x" cos^2 "x" d"x"`.
`int_(pi/5)^((3pi)/10) [(tan x)/(tan x + cot x)]`dx = ?
`int_0^1 x(1 - x)^5 "dx" =` ______.
`int_0^1 sin^-1 ((2x)/(1 + x^2))"d"x` = ______.
Evaluate:
`int (1 + cosx)/(sin^2x)dx`
In Method: Resubstitution, what should be done first?
Which limits are used to evaluate the resulting expression in Method: Resubstitution?
For the integral \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which substitution is suitable?
After using \[t=\tan^{-1}x\] in \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which integral results?
What should not be added in a definite integral?
