Advertisements
Advertisements
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`
Advertisements
उत्तर
Let I = `int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
I = `int_0^(pi/2) cos^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
`2I = int_0^(pi/2) (sin^(3/2)x/(sin^(3/2)x+cos^(3/2) x)+cos^(3/2)x/(sin^(3/2)x + cos^(3/2)x)) dx`
Simplify the numerator:
`(sin^(3/2)x+cos^(3/2) x)/(sin^(3/2)x+cos^(3/2)) = 1`
`2I = int_0^(pi/2) 1 dx`
`int_0^(pi/2) 1 dx = [x]_0^(pi/2)=pi/2 - 0 = pi/2`
`2I = pi/2`
`I=pi/4`
`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`
If `int_0^alpha3x^2dx=8` then the value of α is :
(a) 0
(b) -2
(c) 2
(d) ±2
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
Evaluate : `intsec^nxtanxdx`
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/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
Evaluate `int_1^3 x^2*log x "d"x`
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^4 1/(1 + sqrtx)`dx = ______.
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
`int (dx)/(e^x + e^(-x))` is equal to ______.
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.
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 ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
`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/4) log(1 + tanx)dx`.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
