Advertisements
Advertisements
Question
Evaluate: `int_0^π x/(1 + sinx)dx`.
Solution
`int_0^π x/(1 + sinx)dx`
Let I = `int_0^π x/(1 + sinx)dx` ...(i)
On using property
`int_0^a f(x)dx = int_0^a f(a - x)dx`
∴ I = `int_0^π (π - x)/(1 + sin(π - x))dx`
I = `int_0^π (π - x)/(1 + sinx)dx` ...(ii)
Adding equations (i) and (ii), we get
2I = `int_0^π π/(1 + sinx)dx`
= `πint_0^π 1/(1 + sinx) xx (1 - sinx)/(1 - sinx)dx` ...[Multiplying and dividing by (1 – sin x)]
= `πint_0^π (1 - sinx)/(1 - sin^2x)dx = πint_0^π (1 - sinx)/(cos^2x)dx`
= `π[int_0^π 1/(cos^2x)dx - int_0^π sinx/(cos^2x)dx]`
= `π[int_0^π sec^2x dx - int_0^π secx tanx dx]`
= `π[[tanx]_0^π - [secx]_0^π]`
= π[0 – (– 1 – 1)]
= 2π
∴ I = `(2π)/2` = π.
APPEARS IN
RELATED QUESTIONS
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate`int (1)/(x(3+log x))dx`
`int_0^1 "e"^(2x) "d"x` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)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` = ______
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_a^b f(x)dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
