Advertisements
Advertisements
प्रश्न
Evaluate: `int_0^1 "t"^2 sqrt(1 - "t") "dt"`
Advertisements
उत्तर
Let I = `int_0^1 "t"^2 sqrt(1 - "t") "dt"`
= `int_0^1 (1 -"t")^2 sqrt(1 - (1 - "t")) "dt"` ......`[∵ int_0^"a" "f"(x)"d"x = int_0^"a" "f"("a" - x)"d"x]`
= `int_0^1 (1 - 2"t" + "t"^2)sqrt("t") "dt"`
= `int_0^1("t"^(1/2) - 2"t"^(3/2) + "t"^(5/2))"dt"`
= `int_0^1 "t"^(1/2) "dt" - 2 int_0^1 "t"^(3/2) "dt" + int_0^1 "t"^(5/2) "dt"`
= `[("t"^(3/2))/(3/2)]_0^1 - 2[("t"^(5/2))/(5/2)]_0^1 + [("t"^(7/2))/(7/2)]_0^1`
= `2/3(1^(3/2) - 0) - 4/5(1^(5/2) - 0) + 2/7(1^(7/2) - 0)`
= `2/3 - 4/5 + 2/7`
= `(70 - 84 + 30)/105`
∴ I = `16/105`
APPEARS IN
संबंधित प्रश्न
Evaluate: `int_0^(π/4) cot^2x.dx`
Evaluate: `int_0^oo xe^-x.dx`
Evaluate: `int_0^π sin^3x (1 + 2cosx)(1 + cosx)^2.dx`
Choose the correct option from the given alternatives :
`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` = ______.
Evaluate: `int_(pi/6)^(pi/3) cosx "d"x`
Evaluate: `int_0^1 |x| "d"x`
Evaluate: `int_1^2 x/(1 + x^2) "d"x`
Evaluate: `int_0^1 "e"^x/sqrt("e"^x - 1) "d"x`
Evaluate: `int_(pi/6)^(pi/3) sin^2 x "d"x`
Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x) "d"x`
Evaluate:
`int_0^(pi/2) cos^3x dx`
Evaluate: `int_0^(pi/4) cosx/(4 - sin^2 x) "d"x`
Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`
Evaluate: `int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x) "d"x`
Evaluate: `int_(-1)^1 |5x - 3| "d"x`
Evaluate: `int_(-4)^2 1/(x^2 + 4x + 13) "d"x`
Evaluate: `int_0^1 x* tan^-1x "d"x`
Evaluate: `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2) "d"x`
Evaluate: `int_(-1)^1 1/("a"^2"e"^x + "b"^2"e"^(-x)) "d"x`
Evaluate: `int_0^3 x^2 (3 - x)^(5/2) "d"x`
Evaluate: `int_0^(1/2) 1/((1 - 2x^2) sqrt(1 - x^2)) "d"x`
Evaluate: `int_(1/sqrt(2))^1 (("e"^(cos^-1x))(sin^-1x))/sqrt(1 - x^2) "d"x`
Evaluate: `int_0^1 (log(x + 1))/(x^2 + 1) "d"x`
Evaluate: `int_(-1)^1 (1 + x^2)/(9 - x^2) "d"x`
Evaluate: `int_0^(pi/4) (cos2x)/(1 + cos 2x + sin 2x) "d"x`
Evaluate: `int_0^(pi/4) log(1 + tanx) "d"x`
Evaluate: `int_0^pi 1/(3 + 2sinx + cosx) "d"x`
`int_0^(π/2) sin^6x cos^2x.dx` = ______.
Evaluate: `int_0^1 tan^-1(x/sqrt(1 - x^2))dx`.
Evaluate:
`int_0^(π/2) sin^8x dx`
Evaluate `int_(π/6)^(π/3) cos^2x dx`
Evaluate:
`int_0^(π/2) (sin 2x)/(1 + sin^4x)dx`
Find the value of ‘a’ if `int_2^a (x + 1)dx = 7/2`
Prove that: `int_0^1 logx/sqrt(1 - x^2)dx = π/2 log(1/2)`
