Advertisements
Advertisements
Question
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
`int dx/(e^x + e^(-x))` is equal to ______.
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
Evaluate the following integral:
Evaluate the following integrals as limit of sums:
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums
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_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))` (Hint: Let x = sin θ)
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
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
`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→∞){(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 ______.
`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 ______.
