Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
Advertisements
उत्तर
Let `I = int_0^(pi/2) (cos^5 x)/ (sin^5 x + cos ^5 x) dx` ....(i)
Also, `I = int_0^(pi/2) (cos^5 (pi/2 - x))/(sin^5 (pi/2 - x) + cos^5 (pi/2 - x)) dx`
`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
`= int_0^(pi/2) (sin^5 x)/ (cos^5x + sin^5 x) dx` ....(ii)
Adding (i) and (ii), we have
`2 I = int_0^(pi/2) (cos^5x)/(cos^5x + sin^5 x) dx + int_0^(pi/2) (sin^5x)/ (cos^5 x + sin^5 x) dx`
`= int_0^(pi/2) (cos^5 x + sin^5 x)/ (cos^5 x + sin^5 x) dx`
`= int_0^(pi/2) 1 dx = [x]_0^(pi/2) = pi/2`
Hence, `I = pi/4`
APPEARS IN
संबंधित प्रश्न
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| 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_((-pi)/2)^(pi/2) sin^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 ______.
Evaluate = `int (tan x)/(sec x + tan x)` . dx
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_0^1 x(1 - x)^5 "d"x`
`int_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{pi/2} cos^2x dx` = ______
`int_0^1 x tan^-1x dx` = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^pi x sin^2x dx` = ______
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
`int_0^1 1/(2x + 5) dx` = ______.
`int_a^b f(x)dx` = ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate: `int_0^π x/(1 + sinx)dx`.
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_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
