Advertisements
Advertisements
प्रश्न
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
उत्तर
Let I = `int e^x ((1 - sin x)/(1 + cos x))`dx
`= int e^x [(1 - 2 sin x/2 cos x/2)/(2 sin^2 x/2)]`dx
`= int e^x (1/2 cosec^2 * x/2 - cot x/2)`dx
`= - int cot x/2 e^x dx + 1/2 int e^x cosec^2 x/2 dx`
`= - [cot x/2 * e^x - int - 1/2 cosec^2 x/2 e^x dx] + 1/2 int cosec x/2 * e^x dx`
`= - e^x cot x/2 - 1/2 int cosec^2 x/2 * e^x dx + 1/2 int cosec^2 x/2 * e^x dx`
`= - e^x cot x/2`
`therefore int_(pi/2)^pi e^x ((1 - sin x)/(1 + cos x))`dx
`= - [e^x cot x/2]_(pi/2)^pi`
`= - [pi cot pi/2 - e^(pi/2) cot pi/4]`
`= - 0 + e^(pi/2) * 1`
`= e^(pi/2)`
APPEARS IN
संबंधित प्रश्न
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`
Evaluate the definite integral:
`int_0^(pi/4) (sin x + cos x)/(9+16sin 2x) dx`
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
Prove the following:
`int_0^1 xe^x dx = 1`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
Choose the correct answers The value of `int_0^1 tan^(-1) (2x -1)/(1+x - x^2)` dx is
(A) 1
(B) 0
(C) –1
(D) `pi/4`
Evaluate the following integrals as limit of sums:
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Evaluate:
`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
The value of `lim_(x -> 0) [(d/(dx) int_0^(x^2) sec^2 xdx),(d/(dx) (x sin x))]` is equal to
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"):}`
What is the derivative of `f(x) = |x|` at `x` = 0?
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`lim_(n rightarrow ∞)1/2^n [1/sqrt(1 - 1/2^n) + 1/sqrt(1 - 2/2^n) + 1/sqrt(1 - 3/2^n) + ...... + 1/sqrt(1 - (2^n - 1)/2^n)]` is equal to ______.
