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
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
The total revenue R = 720 - 3x2 where x is number of items sold. Find x for which total revenue R is increasing.
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
Choose the correct alternative:
`int_(-9)^9 x^3/(4 - x^2) "d"x` =
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2} log(tanx)dx` = ______
`int_0^{pi/2} xsinx dx` = ______
`int_0^{pi/2} cos^2x dx` = ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`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)
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
`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.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` 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`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
The area enclosed between the graph of y = x3 and the lines x = 0, y = 1, y = 8 is ______.
