Advertisements
Advertisements
Question
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Advertisements
Solution
Let `I = int (cos x)/sqrt(1 + sin x)` dx
Put 1 + sin x = t
cos x dx = dt
∴ `I = int dt/sqrt(1 + t) = (1 + t)^(1/2 +1)/(1/2 + 1) + C`
`= 2 (1 + t)^(1/2) + C`
`= 2 sqrt(1 + sin x + C)`
APPEARS IN
RELATED QUESTIONS
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`x/(e^(x^2))`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate: ∫ |x| dx if x < 0
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int x^2/sqrt(1 - x^6)` dx = ________________
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int x/(x + 2) "d"x`
`int(log(logx))/x "d"x`
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int secx/(secx - tanx)dx` equals ______.
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
After putting \[t=\cos x\], which integral is obtained from \[\int\sin^2x\cos^2x(\sin x)\,dx\]?
