Advertisements
Advertisements
Question
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Advertisements
Solution
Let `I = int sqrtsin 2x cos 2x dx`
Put sin 2x = t
⇒ 2 cos 2x dx = dt
∴ `I = 1/2 int t^(1/2) dt = 1/2 * t^(1/2 + 1)/(1/2 + 1) + C`
`1/2 xx 2/3 t^(3/2) + C = 1/2 t^(3/2) + C`
`1/3 (sin 2x)^(3/2) + C`
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`1/(1 + cot x)`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
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
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
`int logx/(log ex)^2*dx` = ______.
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: ∫ |x| dx if x < 0
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int 1/(xsin^2(logx)) "d"x`
`int(1 - x)^(-2) dx` = ______.
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int sec^6 x tan x "d"x` = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate `int1/(x(x - 1))dx`
`int x^3 e^(x^2) dx`
`int x^2/sqrt(1 - x^6)dx` = ______.
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
