Advertisements
Advertisements
प्रश्न
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Advertisements
उत्तर
Let I = `int_0^(π/4) log_e (1 + tan x)dx` ...(i)
`\implies` I = `int_0^(π/4) log_e (1 + tan(π/4 - x))dx`,
Using `int_0^a f(x)dx = int_0^a f(a - x)dx`
`\implies` I = `int_0^(π/4) log_e (1 + (1 - tanx)/(1 + tanx))dx`
= `int_0^(π/4) log_e (2/(1 + tanx))dx`
= `int_0^(π/4) log_e 2dx - I` ...(Using ...(i))
`\implies` 2I = `π/4 log_e 2`
`\implies` I = `π/8 log_e 2`.
APPEARS IN
संबंधित प्रश्न
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
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^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_1^2 1/(2x + 3) dx` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
`int_4^9 1/sqrt(x)dx` = ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
