Advertisements
Advertisements
Question
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Advertisements
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 `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_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
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 ______.
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`
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integral:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
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
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
What is the derivative of `f(x) = |x|` at `x` = 0?
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.
