Advertisements
Advertisements
प्रश्न
Evaluate the integral by using substitution.
`int_0^2 xsqrt(x+2)` (Put x + 2 = `t^2`)
Advertisements
उत्तर
Let `I = int_0^2 x sqrt (x + 2) dx`
Put x + 2 = t
⇒ dx = dt
When x = 0, t = 2 and when x = 2, t = 4
∴ `I = int_2^4 (t - 2) sqrtt dt `
`= int_2^4 (t^(3/2) - 2t^(1/2)) dt`
`= [2/5 t^(5/2) - 2 xx 2/3 t^(3/2)]_2^4`
`= [2/5 (4)^(5/2) - 4/3 t^(3/2)]_2^4`
`= [2/5 (4)^(5/2) - 4/3 (4)^(3/2)] - [2/5 (2)^(5/2) = 4/3 (2)^(3/2)]`
`= 2/5 (2)^5 - 4/3 (2)^3 - 2/5 xx 4sqrt2 + 4/3 xx 2sqrt2`
`= 2/5 xx 32 - 4/3 xx 8 - 8/5 sqrt2 + 8/3 sqrt2`
`= 64/5 - 32/3 - (8/5 sqrt2 - 8/3 sqrt2)`
`= (192 - 160)/15 - ((24sqrt2 - 40sqrt2))/15`
`= 32/15 + (16sqrt2)/15`
`= 16/15 (2+sqrt2)`
or `(16sqrt2)/15 (sqrt2+1)`
APPEARS IN
संबंधित प्रश्न
Evaluate `∫_0^(3/2)|x cosπx|dx`
Evaluate `int_(-1)^2|x^3-x|dx`
find `∫_2^4 x/(x^2 + 1)dx`
Evaluate the integral by using substitution.
`int_0^1 x/(x^2 +1)`dx
Evaluate the integral by using substitution.
`int_0^(pi/2) sqrt(sin phi) cos^5 phidphi`
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_1^2 (1/x- 1/(2x^2))e^(2x) dx`
`int 1/(1 + cos x)` dx = _____
A) `tan(x/2) + c`
B) `2 tan (x/2) + c`
C) -`cot (x/2) + c`
D) -2 `cot (x/2)` + c
Evaluate of the following integral:
(i) \[\int x^4 dx\]
Evaluate:
Evaluate:
Evaluate the following integral:
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 :
Evaluate: \[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx\] .
Evaluate: `int_ e^x ((2+sin2x))/cos^2 x dx`
Find: `int_ (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.
Evaluate the following:
`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`
The value of `int_0^1 (x^4(1 - x)^4)/(1 + x^2) dx` is
Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.
If `int x^5 cos (x^6)dx = k sin (x^6) + C`, find the value of k.
Which limits are used to evaluate the resulting expression in Method: Resubstitution?
Under Method: Changing the Limits, what does the lower limit \[a\] become?
After changing the limits to the new variable, how is the definite integral completed?
For the integral \[\int_{0}^{1}\frac{\tan^{-1}x}{1+x^2}\,dx\], which substitution is suitable?
If \[t=\tan^{-1}x\], what is \[dt\]?
