Advertisements
Advertisements
Question
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
Solution
Let I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x).dx` ...(i)
= `int_2^5 (sqrt(2 + 5 - x))/(sqrt(2 + 5 - x) + sqrt(7 - (2 + 5 - x))).dx` ...`[∵ int_a^b f(x)dx = int_a^b f(a + b - x)dx]`
∴ I = `int_2^5 (sqrt(7 - x))/(sqrt(7 - x) + sqrt(x)).dx` ...(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x).dx + int_2^5 sqrt(7 - x)/(sqrt(7 - x) + sqrt(x)).dx`
= `int_2^5 (sqrt(x) + sqrt(7 - x))/(sqrt(x) + sqrt(7 - x)).dx`
= `int_2^5 1.dx`
= `[x]_2^5`
∴ 2I = 5 – 2
2I = 3
I = `3/2`
APPEARS IN
RELATED QUESTIONS
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_(-a)^asqrt((a-x)/(a+x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan 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_((-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)`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
Evaluate : ∫ log (1 + x2) dx
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_0^2 e^x dx` = ______.
`int_0^1 "e"^(2x) "d"x` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x`.
Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x` ......(i)
Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`, we get
I = `int_2^5 ("( )")/(sqrt(7 - x) + "( )") "d"x` ......(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x)) "d"x + ( ) "d"x`
2I = `int_2^5 (("( )" + "( )")/("( )" + "( )")) "d"x`
2I = `square`
∴ I = `square`
`int_0^1 (1 - x)^5`dx = ______.
`int_0^1 x tan^-1x dx` = ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_a^b f(x)dx` = ______.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
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.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following definite integral:
`int_1^3 log 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: `int_-1^1 x^17.cos^4x dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
