Advertisements
Advertisements
Question
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Advertisements
Solution
Let f(x) = `log((2 - x)/(2 + x))`
We have, f(– x) = `log((2 + x)/(2 - x))`
= `-log((2 - x)/(2 + x))`
= – f(x)
So, f(x) is an odd function.
∴ `int_-1^1 log ((2 - x)/(2 + x))dx` = 0.
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_0^(pi/2) (2log sin x - log sin 2x)dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
Evaluate the following integral:
`int_0^1 x(1 - x)^5 *dx`
`int_0^2 e^x dx` = ______.
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate `int_0^(π//4) log (1 + tanx)dx`.
Evaluate `int_-1^1 |x^4 - x|dx`.
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
