Advertisements
Advertisements
Question
Advertisements
Solution
\[\int\frac{\sin x}{\left( 1 + \cos x \right)^2}dx\]
\[\text{Let 1} + \cos x = t\]
\[ \Rightarrow - \sin x = \frac{dt}{dx}\]
\[ \Rightarrow \text{sin x dx} = - dt\]
\[Now, \int\frac{\sin x}{\left( 1 + \cos x \right)^2}dx\]
\[ = \int - \frac{dt}{t^2}\]
\[ = - \int t^{- 2} dt\]
\[ = - \left[ \frac{t^{- 2 + 1}}{- 2 + 1} \right] + C\]
\[ = \frac{1}{t} + C\]
\[ = \frac{1}{1 + \cos x} + C\]
APPEARS IN
RELATED QUESTIONS
Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums
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_(-1)^1 e^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/4) (sin x + cos x)/(9+16sin 2x) dx`
Evaluate the definite integral:
`int_0^(pi/2) sin 2x tan^(-1) (sinx) dx`
Prove the following:
`int_0^1 xe^x dx = 1`
Prove the following:
`int_(-1)^1 x^17 cos^4 xdx = 0`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
Evaluate `int_0^1 e^(2-3x) dx` as a limit of a sum.
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_1^4 ( 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 (x^2 + 3) "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
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 limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
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 ______.
`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 ______.
