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 `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 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 following definite integrals as limit of sums.
`int_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 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`
Evaluate the definite integral:
`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
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_0^2 ("d"x)/("e"^x + "e"^-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
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 ______.
`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 ______.
