Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
उत्तर
Let I = `int_((-pi)/2)^(pi/2) log((2 + sin x)/(2 - sin x)) * dx`
Let f(x) = `log((2 + sin x)/(2 - sin x))`
∴ f(– x)= `log[(2 + sin (-x))/(2 - sin (-x))]`
= `log((2 - sin x)/(2 + sin x))`
= `-log((2 + sin x)/(2 + sin x))`
= – f(x)
∴ f is an odd function.
∴ `int_((-pi)/2)^(pi/2) f(x) * dx` = 0
∴ `int_((-pi)/2)^(pi/2)log((2 + sin x)/(2 - sin x)) * dx` = 0.
APPEARS IN
संबंधित प्रश्न
Evaluate:
`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`
Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`
Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
Evaluate the following : `int_0^1 (logx)/sqrt(1 - x^2)*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 :
If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then
Evaluate the following : `int_0^(pi/2) cosx/(3cosx + sinx)*dx`
Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`
Evaluate the following:
`int_0^pi x/(1 + sin^2x) * dx`
Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`
Evaluate the following : `int_0^a 1/(a^2 + ax - x^2)*dx`
Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`
Evaluate the following : `int_(-2)^(3) |x - 2|*dx`
Evaluate the following : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k
Choose the correct alternative :
`int_0^2 e^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_(-5)^(5) x^3/(x^2 + 7)*dx` = 0
State whether the following is True or False : `int_4^7 ((11 - x)^2)/((11 - x)^2 + x^2)*dx = (3)/(2)`
Solve the following : `int_2^3 x/(x^2 + 1)*dx`
Solve the following : `int_1^2 x^2*dx`
Solve the following : `int_2^4 x/(x^2 + 1)*dx`
State whether the following statement is True or False:
`int_2^3 x/(x^2 + 1) "d"x = 1/2 log 2`
Evaluate `int_0^1 1/(sqrt(1 + x) + sqrt(x)) "d"x`
Evaluate `int_1^2 (3x)/((9x^2 - 1)) "d"x`
Evaluate the following definite integrals:
`int _1^2 (3x) / ( (9 x^2 - 1)) * dx`
Evaluate the following definite integrals: `int_-2^3 1/(x + 5) *dx`
Evaluate:
`int_(-π/2)^(π/2) (sin^3x)/(1 + cos^2x)dx`
Solve the following:
`int_1^3 x^2 log x dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Solve the following.
`int_1^3x^2logx dx`
Evaluate the following definite integral:
`int_1^3 logx.dx`
Evaluate the following definite intergral:
`int_4^9 1/sqrtx dx`
Solve the following.
`int_1^3 x^2 logxdx`
Evaluate the following definite intergral:
`int_-2^3 1/(x+5)dx`
Evaluate the following definite intergral:
`int_(-2)^3 1/(x + 5)dx`
Evaluate the following definite intergral:
`int_1^3 log x dx`
Evaluate the following definite intergral:
`int_4^9(1)/sqrtxdx`
