Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\int_0^\pi \frac{\sin x}{\sin x + \cos x} d x\]
\[ = \frac{1}{2} \int_0^\pi \frac{2\sin x}{\sin x + \cos x} d x\]
\[ = \frac{1}{2} \int_0^\pi \frac{\left( \sin x + \cos x \right) - \left( \cos x - \sin x \right)}{\sin x + \cos x} d x\]
\[ = \frac{1}{2} \int_0^\pi dx - \frac{1}{2} \int_0^\pi \frac{\cos x - \sin x}{\sin x + \cos x}dx\]
\[ = \frac{1}{2} \left[ x \right]_0^\pi - \frac{1}{2} \left[ \log\left| \sin x + \cos x \right| \right]_0^\pi \]
\[ = \frac{1}{2}\left[ \pi - 0 \right] - \frac{1}{2}\left[ \log1 - \log1 \right]\]
\[ = \frac{\pi}{2}\]
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 definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos 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`
Prove the following:
`int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3`
Prove the following:
`int_0^(pi/2) sin^3 xdx = 2/3`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
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 : `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_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate:
`int (sin"x"+cos"x")/(sqrt(9+16sin2"x")) "dx"`
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:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
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)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
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
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_(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 ______.
