Advertisements
Advertisements
प्रश्न
Evaluate:
`int_0^1 sqrt((1 - x)/(1 + x)) * dx`
उत्तर
Let `I = int_0^1 sqrt((1 - x)/(1 + x)) * dx`
Put x = cos θ
dx = − sinθ dθ
When x = 0, cos θ = 0 = cos `pi/(2)` ∴ θ = `pi/(2)`
When x = 1, cos θ = 1 = cos 0 ∴ θ = 0
∴ `I = int_(pi/2)^0 sqrt(( - cos theta)/(1 + cos theta)) * (- sin θ) dθ`
= `int_(pi/2)^0 sqrt((2sin^2(theta//2))/(2cos^2(theta//2)))(- 2sin theta/2 cos theta/2) * dθ`
= `int_(pi/2)^0 (sin(theta//2)/(cos(theta//2)))[- 2sin (theta/2) cos (theta/2)] * dθ`
= `int_(pi/2)^0 - 2sin^2(theta/2) * dθ`
= `- int_(pi/2)^0 (1 - cos θ) * dθ`
= `-[theta - sintheta]_(pi/2)^0`
= `-[(0 - sin0) - (pi/2 - sin pi/2)]`
= `-[0 - pi/2 + 1]`
= `pi/(2) - 1`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int 1/(a^2 - x^2) dx = 1/2 a log ((a +x)/(a-x)) + c`
Evaluate : `int_0^(pi/4) sin 4x sin 3x *dx`
Evaluate : `int_0^(pi/4) sec^4x*dx`
Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`
Choose the correct option from the given alternatives :
`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Evaluate the following : `int_(-1)^(1) (1 + x^3)/(9 - x^2)*dx`
Evaluate the following : `int_0^a 1/(a^2 + ax - x^2)*dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
Choose the correct alternative :
`int_(-2)^3 dx/(x + 5)` =
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_2^3 x^4*dx` = _______
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_"a"^"b" f(x - "a" - "b")*dx`
State whether the following is True or False : `int_1^2 sqrt(x)/(sqrt(3 - x) + sqrt(x))*dx = (1)/(2)`
Solve the following : `int_0^1 (1)/(sqrt(1 + x) + sqrt(x))dx`
Solve the following : `int_1^2 (5x^2)/(x^2 + 4x + 3)*dx`
Solve the following : `int_1^2 dx/(x(1 + logx)^2`
`int_0^1 sqrt((1 - x)/(1 + x)) "d"x` =
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
Prove that: `int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`. Hence find `int_0^(pi/2) sin^2x "d"x`
Choose the correct alternative:
`int_0^"a" 3x^5 "d"x` = 8, then a =
State whether the following statement is True or False:
`int_0^"a" 3x^2 "d"x` = 27, then a = 2.5
`int_((-pi)/8)^(pi/8) log ((2 - sin x)/(2 + sin x))` dx = ______.
`int_0^(pi/2) (cos x)/((4 + sin x)(3 + sin x))`dx = ?
Evaluate the following definite integrals:
`int _1^2 (3x) / ( (9 x^2 - 1)) * dx`
`int_0^(π/2) (sin^2 x.dx)/(1 + cosx)^2` = ______.
Evaluate:
`int_(-π/2)^(π/2) (sin^3x)/(1 + cos^2x)dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5) *dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Solve the following:
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following definite intergral:
`int_4^9(1)/sqrtxdx`
Evaluate the following definite intergral:
`int_1^2(3x)/(9x^2-1).dx`
Evaluate the following definite integrals: `int_1^2 (3x)/((9x^2 - 1))*dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite intergral.
`int_4^9 1/sqrtx .dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2) dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
