Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Advertisements
उत्तर
Let I = `int_0^1 (x"d"x)/sqrt(1 + x^2)`
Put 1 + x2 = t
⇒ 2x dx = dt
⇒ x dx = `"dt"/2`
Changing the limits, we have
When x = 0
∴ t = 1
When x = 1
∴ t = 2
∴ I = `1/2 int_1^2 "dt"/sqrt("t")`
= `1/2 * 2["t"^(1/2)]_1^2`
= `sqrt(2) - 1`
Hence, I = `sqrt(2) - 1`.
APPEARS IN
संबंधित प्रश्न
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3`
Prove the following:
`int_0^1 xe^x dx = 1`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
`int dx/(e^x + e^(-x))` is equal to ______.
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Evaluate:
`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`
If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
If f" = C, C ≠ 0, where C is a constant, then the value of `lim_(x -> 0) (f(x) - 2f (2x) + 3f (3x))/x^2` is
Left `f(x) = {{:(1",", "if x is rational number"),(0",", "if x is irrational number"):}`. The value `fof (sqrt(3))` is
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
