Advertisements
Advertisements
प्रश्न
Evaluate : `int_0^(pi/4) sin^4x*dx`
उत्तर
Consider sin4x = `(sin^2x)^2`
= `((1 - cos 2x)/2)^2`
= `(1)/(4)[1 - 2 cos 2x + cos^2 2x]`
= `(1)/(4)[1 - 2 cos 2x + (1 + cos 4x)/2]`
= `(1)/(4)[3/2 - 2 cos 2x + 1/2 cos 4x]`
∴ `int_0^(pi/4) sin^4x*dx`
= `(1)/(4) int_0^(pi/4) [3/2 - 2 cos 2x 1/2 cos 4x]*dx`
= `(3)/(8) int_0^(pi/4) 1*dx - 1/2 int_0^(pi/4) cos 2x*dx + 1/8 int_0^(pi/4) cos 4x*dx`
= `(3)/(8)[x]_0^(pi/4) - 1/2[(sin2x)/2]_0^(pi/4) + 1/8[(sin4x)/4]_0^(pi/4)`
= `(3)/(8)[pi/4 - 0] - 1/4[sin pi/2 - sin0] + 1/32[sin pi - sin0]`
= `(3pi)/(32) - (1)/(4)[1 - 0] + (1)/(32)(0 - 0)`
= `(3pi)/(32) - (1)/(4)`
= `(3pi - 8)/(32)`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_(-4)^2 (1)/(x^2 + 4x + 13)*dx`
Evaluate : `int_0^1 x tan^-1x*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3) * dx` = ______.
Choose the correct option from the given alternatives :
If `dx/(sqrt(1 + x) - sqrt(x)) = k/(3)`, then k is equal to
Choose the correct option from the given alternatives :
`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following definite integral:
`int_(-2)^3 (1)/(x + 5)*dx`
Evaluate the following definite integrals: `int_0^1 (x^2 + 3x + 2)/sqrt(x)dx`
Evaluate the following definite integrals: `int_2^3 x/((x + 2)(x + 3)). dx`
Evaluate the following definite integrals: If `int_0^"a" (2x + 1)*dx` = 2, find the real value of a.
Evaluate the following integrals : `int_0^"a" x^2("a" - x)^(3/2)*dx`
Choose the correct alternative :
`int_4^9 dx/sqrt(x)` =
Choose the correct alternative :
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =
Fill in the blank : `int_2^3 x/(x^2 - 1)*dx` = _______
Fill in the blank : `int_(-2)^3 dx/(x + 5)` = _______
State whether the following is True or False : `int_"a"^"b" f(x)*dx = int_"a"^"b" f(x - "a" - "b")*dx`
Solve the following : `int_(-2)^3 (1)/(x + 5)*dx`
`int_1^9 (x + 1)/sqrt(x) "d"x` =
Choose the correct alternative:
`int_4^9 ("d"x)/sqrt(x)` =
Choose the correct alternative:
`int_2^3 x/(x^2 - 1) "d"x` =
State whether the following statement is True or False:
`int_0^"a" 3x^2 "d"x` = 27, then a = 2.5
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
Evaluate `int_1^2 "e"^(2x) (1/x - 1/(2x^2)) "d"x`
`int_0^1 tan^-1 ((2x - 1)/(1 + x - x^2))` dx = ?
`int_2^3 "x"/("x"^2 - 1)` dx = ____________.
Evaluate the following definite integral :
`int_1^2 (3"x")/((9"x"^2 - 1)) "dx"`
Evaluate the following integrals:
`int_0^1 x(1 - x)^5 dx`
Solve the following `int_1^3 x^2log x dx`
Evaluate the following definite integral:
`int_-2^3 1/(x+5) *dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
Evaluate the following definite integral:
`int_1^3 logx dx`
Evaluate the following definite integral:
`int_1^2 (3x)/((9x^2 - 1)) dx`
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 definite intergral:
`int_1^3 log x dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
