Advertisements
Advertisements
Question
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Solution
Let `I=int_0^pi(xsinx)/(1+sinx)dx`
`=int_0^pi((pi-x)sin(pi-x))/(1+sin(pi-x))dx [because int_0^a f(x)dx=int_0^af(a-x)dx]`
`=int_0^pi((pi-x)sinx)/(1+sinx)dx`
`=int_0^pi(pisinx)/(1+sinx)dx-I`
`I=int_0^pi(pisinx)/(1+sinx)dx-I`
`2I=int_0^pi(pisinx.(1-sinx))/((1+sinx)(1-sinx))dx`
`2I=int_0^pi(pisinx.(1-sinx))/(1-sin^2x)dx`
`(2I)/pi=int_0^pi(sinx.(1-sinx))/cos^2xdx`
`(2I)/pi=int_0^pi(sinx.-sin^2x)/cos^2xdx`
`(2I)/pi=int_0^pi(sinx)/cos^2xdx-int_0^pi(sin^2x)/cos^2xdx`
`(2I)/pi=int_0^pisecx.tanxdx-int_0^pitan^2xdx`
`(2I)/pi=[secx]_0^pi-int_0^pi(sec^2x-1)dx`
`(2I)/pi=[secpi-sec0]-int_0^pisec^2x.dx+int_0^pi1dx`
`(2I)/pi=[-1-1]-[tanx]_0^pi_[x]_0^pi`
`(2I)/pi=[-2]-[tanpi-tan0]+pi`
`(2I)/pi=[-2]-0+pi`
`thereforeI=((pi-2)pi)/2`
APPEARS IN
RELATED QUESTIONS
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_1^2 1/(2x + 3) dx` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`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^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_0^9 1/(1 + sqrtx)` dx = ______
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_4^9 1/sqrt(x)dx` = ______.
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
