Advertisements
Advertisements
Question
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Solution
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
RELATED QUESTIONS
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_1^4 (x^2 - x) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
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`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
Choose the correct answers The value of `int_0^1 tan^(-1) (2x -1)/(1+x - x^2)` dx is
(A) 1
(B) 0
(C) –1
(D) `pi/4`
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`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 ______.
