Advertisements
Advertisements
Question
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Solution
We have I = `int_0^(pi/2) (sin^2x)/(sinx + cosx) "d"x`
= `int_0^(pi/2) (sin^2(pi/2 - x))/(sin(pi/2 - x) + cos(pi/2 - x)) "d"x` ....(By P4)
⇒ I = `int_0^(pi/2) (cos^2x)/(sinx + cosx) "d"x`
Thus, we get 2I = `1/sqrt(2) int_0^(pi/2) ("d"x)/(cos(x - pi/4))`
= `1/sqrt(2) int_0^(pi/2) sec(x - pi/2) "d"x`
= `1/sqrt(2) [log(sec(x - pi/4) + tan(x - pi/4))]_0^(pi/2)`
= `1/sqrt(2)[log(sec pi/4 + tan pi/4) - log sec(- pi/4) + tan(- pi/4)]`
= `1/sqrt(2) [log(sqrt(2) + 1) - log(sqrt(2) - 1)]`
= `1/sqrt(2) log|(sqrt(2) + 1)/(sqrt(2) - 1)|`
= `1/sqrt(2) log((sqrt(2) - 1)^2/1)`
= `2/sqrt(2) log(sqrt(2) + 1)`
Hence I = `1/sqrt(2) log(sqrt(2) + 1)`.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^pi sin^2x.cos^2x dx` = ______
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate:
`int_0^6 |x + 3|dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
