Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
उत्तर
Let I = `int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
= `int_0^(pi/2) (sinx/cosx)/(1 + "m"^2 (sin^2x)/(cos^2x)) "d"x`
= `int_0^(pi/2) (sinx/cosx)/((cos^2x + "m"^2 sin^2x)/cos^2x) "d"x`
= `int_0^(pi/2) (sin x cos x)/(cos^2x + "m"^2 sin^2x) "d"x`
= `int_0^(pi/2) (sinx cosx)/(1 - sin^2x + "m"^2 sin^2x) "d"x`
= `int_0^(pi/2) (sinx cosx)/(1 - sin^2x (1 - "m"^2)) "d"x`
Put sin2x = t
2 sin x cos x dx = dt
sin x cos x dx = `"dt"//2`
Changing the limits we get,
When x = 0
∴ t = sin20 = 0
When x = `pi/2`
∴ t = `sin^2 pi/2` = 1
∴ I = `1/2 int_0^1 "dt"/(1 - (1 - "m"^2)"t")`
I = `1/2 int_0^1 "dt"/(1 + ("m"^2 - 1)"t")`
= `1/2 [(log [1 + "m"^2 - 1)"t")/("m"^2 - 1)]_0^1`
= `1/(2("m"^2 - 1)) [log(1 + "m"^2 - 1) - log(1)]`
= `(log|"m"^2|)/(2("m"^2 - 1))`
Hence, I = `(log|"m"^2|)/(2("m"^2 - 1)) = (log|"m"|)/("m"^2 - 1)`.
APPEARS IN
संबंधित प्रश्न
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.
(A) `1/2`
(B) `1/3`
(C) `1/4`
(D) `1/5`
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
