Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{Let I} = \int e^\ cos^2 x \sin2x dx\]
\[ Let \cos^2 x = t\]
\[ \text{On differentiating both sides, we get}\]
\[ - \text{2 }\text{cos x sin x dx} = dt\]
\[ \therefore I = \int e^t 2 \sin x \cos x \frac{dt}{- 2 \sin x \cos x}\]
\[ = - \int e^t dt\]
\[ = - e^t + c\]
\[ = - e^\ cos^2 x + c\]
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_0^5 (x+1) 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_0^4 (x + e^(2x)) dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos 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_0^1 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_0^1 xe^x dx = 1`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
`int (cos 2x)/(sin x + cos x)^2dx` 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.
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
Evaluate the following integral:
Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x
)`
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "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)
Evaluate the following:
`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2) "d"x`
What is the derivative of `f(x) = |x|` at `x` = 0?
