Advertisements
Advertisements
प्रश्न
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intlogx/(1+logx)^2dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_1^2 1/(2x + 3) dx` = ______
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
`int_a^b f(x)dx` = ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
`int_0^1|3x - 1|dx` equals ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
Which property is useful for piecewise functions or modulus (absolute value) functions?
Which formula is \[P_5\] : Halving the Upper Limit (Addition)?
For an Even Function, which condition is satisfied?
If \[f(-x)=-f(x)\], what is \[\int_{-a}^{a} f(x)\,dx\]?
Which expression is equal to \[\frac{1}{1+\sqrt{\tan x}}\] in the evaluation of \[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\]?
If \[I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\], what is the value of \[I\]?
