Advertisements
Advertisements
प्रश्न
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Advertisements
उत्तर
\[\text{We have}, \]
\[ \int_0^k \frac{1}{2 + 8 x^2} d x = \frac{\pi}{16}\]
\[ \Rightarrow \frac{1}{8} \int_0^k \frac{1}{\frac{1}{4} + x^2} d x = \frac{\pi}{16}\]
\[ \Rightarrow \frac{1}{4} \left[ \tan^{- 1} 2x \right]_0^k = \frac{\pi}{16}\]
\[ \Rightarrow \tan^{- 1} 2k = \frac{\pi}{4}\]
\[ \Rightarrow 2k = \tan\frac{\pi}{4}\]
\[ \Rightarrow 2k = 1\]
\[ \Rightarrow k = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^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/2) (sin x - cos x)/(1+sinx cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) 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.
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "dx"`
Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_1^2 1/(2x + 3) dx` = ______
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2} xsinx dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_(-1)^1 log ((2 - x)/(2 + x)) "dx" = ?`
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`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:
`int_0^1 1/(2x + 5) dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.
`int_(π/3)^(π/2) x sin(π[x] - x)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 `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
Which formula expresses \[P_1\] : Reversing Limits?
