Advertisements
Advertisements
Question
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Advertisements
Solution
Let I = `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx` ...(i)
I = `int_0^(π/2) 1/(1 + [tan(π/2 - x)]^(2/3)) dx` ...[Using property `int_0^a f(x)dx = int_0^a f(a - x)dx`]
I = `int_0^(π/2) 1/(1 + (cot x)^(2/3)) dx`
I = `int_0^(π/2) ((tanx)^(2/3))/((tanx)^(2/3) + 1) dx`
I = `int_0^(pi/2) ((tanx)^(2/3) + 1 - 1)/((tanx)^(2/3) + 1) dx`
I = `int_0^(π/2) (1 + (tanx)^(3/2))/(1 + (tanx)^(3/2)) dx - int_0^(π/2) 1/(1 + (tanx)^(3/2)) dx`
I = `int_0^(π/2) 1.dx - I` ...[From equation (i)]
2I = `int_0^(π/2) 1.dx`
2I = `[x]_0^(π/2)`
2I = `π/2`
I = `π/4`
APPEARS IN
RELATED QUESTIONS
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate = `int (tan x)/(sec x + tan x)` . dx
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_"a"^"b" "f"(x) "d"x` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
`int_(-5)^5 x^7/(x^4 + 10) dx` = ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_0^1 1/(2x + 5) dx` = ______.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
`int_1^2 x logx dx`= ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
