Advertisements
Advertisements
Question
Evaluate the integral by using substitution.
`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate: `int (1+logx)/(x(2+logx)(3+logx))dx`
find `∫_2^4 x/(x^2 + 1)dx`
If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.
Evaluate the integral by using substitution.
`int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx`
Evaluate the integral by using substitution.
`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`
Evaluate the integral by using substitution.
`int_(-1)^1 dx/(x^2 + 2x + 5)`
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 the following integral:
Evaluate the following integral:
Evaluate
\[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\]
Evaluate the following integral:
Find : \[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\] .
Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .
Evaluate: `int_-1^2 (|"x"|)/"x"d"x"`.
Evaluate: `int_1^5{|"x"-1|+|"x"-2|+|"x"-3|}d"x"`.
`int_0^1 sin^-1 ((2x)/(1 + x^2))"d"x` = ______.
Evaluate the following:
`int "dt"/sqrt(3"t" - 2"t"^2)`
`int_0^1 x^2e^x dx` = ______.
