Advertisements
Advertisements
प्रश्न
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`.
विकल्प
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true, but (R) is not the correct explanation of (A).
(A) is true, but (R) is false.
(A) is false, but (R) is true.
Advertisements
उत्तर
Both (A) and (R) are true and (R) is the correct explanation of (A).
Explanation:
I = `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` ...(i)
Using property of definite integral
`int_a^b f(x) dx = int_a^b f(a + b - x) dx`
I = `int_2^8 sqrt(x)/(sqrt(10 - x) + sqrt(x))dx` ...(ii)
Adding equations (i) and (ii)
2I = `int_2^8 (sqrt(10 - x) + sqrt(x))/(sqrt(10 - x) + sqrt(x))dx`
= `int_2^8 dx`
= `[x]_2^8`
= 8 – 2
= 6
`\implies` I = 3
R is also true as the property P4 is
`int_a^b f(x)dx = int_a^b f(a + b - x)`
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_0^2 xsqrt(2 -x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
`int_0^2 e^x dx` = ______.
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate: `int_-1^1 x^17.cos^4x 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`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
Why can \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\] be written as \[2\int_{0}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?
