Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
Evaluate:
`int_0^(pi/4) log (1+ tan x) dx`
Advertisements
उत्तर
Let I = `int_0^(pi/4) log (1 + tan x) dx` ....(1)
∴ I = `int_0^(pi/4) log [1 + tan (pi/4 - x)] dx` `...[int_0^a f(x) dx = int_0^a f(a - x) dx]`
⇒ I = `int_0^(pi/4) log {1 + (tan pi/4 - tan x)/(1 + tan pi/4 tan x)}dx`
⇒ I = `int_0^(pi/4) log {1 + (1 - tan x)/(1 + tan x)} dx`
⇒ I = `int_0^(pi/4) log 2/((1 + tan x)) dx`
⇒ I = `int_0^(pi/4) log 2 dx - int_0^(pi/4) log (1 + tan x) dx`
⇒ I = `int_0^(pi/4) log 2 dx - I` ...[From (1)]
⇒ 2I = `[x log 2]_0^(pi/4)`
⇒ 2I = `pi/4 log 2`
⇒ I = `pi/8 log 2`
APPEARS IN
संबंधित प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
`int_"a"^"b" "f"(x) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
Which of the following is true?
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_(-2)^2 |x cos pix| "d"x` is equal to ______.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
`int_a^b f(x)dx` = ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
