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: `int (1+logx)/(x(2+logx)(3+logx))dx`
Evaluate : `int1/(3+5cosx)dx`
If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.
Evaluate :
`int_e^(e^2) dx/(xlogx)`
Evaluate: `intsinsqrtx/sqrtxdx`
Evaluate the integral by using substitution.
`int_0^1 x/(x^2 +1)`dx
Evaluate the integral by using substitution.
`int_0^1 sin^(-1) ((2x)/(1+ x^2)) 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 of the following integral:
Evaluate:
Evaluate the following integral:
Evaluate the following integral:
\[\int\limits_0^2 \left| x^2 - 3x + 2 \right| dx\]
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
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 each of the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate : \[\int\limits_{- 2}^1 \left| x^3 - x \right|dx\] .
Find : \[\int e^{2x} \sin \left( 3x + 1 \right) dx\] .
Evaluate: `int_1^5{|"x"-1|+|"x"-2|+|"x"-3|}d"x"`.
If `I_n = int_0^(pi/4) tan^n theta "d"theta " then " I_8 + I_6` equals ______.
`int_0^3 1/sqrt(3x - x^2)"d"x` = ______.
Evaluate the following:
`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`
Evaluate the following:
`int "dt"/sqrt(3"t" - 2"t"^2)`
