Advertisements
Advertisements
प्रश्न
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Advertisements
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_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/6)^(pi/3) (sin x + cosx)/sqrt(sin 2x) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sin x + cos x)/(9+16sin 2x) dx`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
If f (a + b - x) = f (x), then `int_a^b x f(x )dx` is equal to ______.
if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.
(A) `1/2`
(B) `1/3`
(C) `1/4`
(D) `1/5`
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
Evaluate the following integral:
Evaluate the following integrals as limit of sums:
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
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 `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to
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 ______.
