Advertisements
Advertisements
प्रश्न
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Advertisements
उत्तर
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = `1/2`.
Explanation:
Given that `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`
⇒ `1/4 int_0^"a" 1/((1/4 + x^2)) "d"x = pi/8`
⇒ `int_0^pi 1/([(1/2)^2 + x^2]) "d"x = pi/2`
⇒ `1/(1/2) [tan^-1 x/(1/2)]_0^"a" = pi/2`
⇒ `2[tan^-1 2"a" - tan^-1 0] = pi/2`
⇒ `tan^-1 2"a" = pi/4`
⇒ 2a = `tan pi/4`
⇒ 2a = 1
⇒ a = `1/2`.
APPEARS IN
संबंधित प्रश्न
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_0^2 e^x dx` = ______.
`int_"a"^"b" "f"(x) "d"x` = ______
`int_2^4 x/(x^2 + 1) "d"x` = ______
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_(-pi/4)^(pi/4) 1/(1 - sinx) "d"x` = ______.
`int_0^(pi/2) 1/(1 + cos^3x) "d"x` = ______.
`int_0^1 "e"^(5logx) "d"x` = ______.
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
Solve the following.
`int_1^3 x^2 logx dx`
`int_1^2 x logx dx`= ______
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
Under which condition does \[P_6\] give \[\int_{0}^{2a} f(x)\,dx=2\int_{0}^{a} f(x)\,dx\]?
Under which condition does \[P_6\] give \[\int_{0}^{2a} f(x)\,dx=0\]?
If \[f(-x)=-f(x)\], what is \[\int_{-a}^{a} f(x)\,dx\]?
