Advertisements
Advertisements
Question
Integrate the functions:
sin (ax + b) cos (ax + b)
Advertisements
Solution
Let `I = int sin (ax + b) cos (ax + b) dx`
Put sin (ax + b) = t
⇒ a cos (ax + b) dx = dt
∴ `I = 1/a int t dt = 1/a * t^2/2 + C`
`= 1/(2a) t^2 + C`
`= 1/ (2a) sin^2 (ax + b) + C`
Or, put cos (ax + b) = t
⇒ -a sin (ax + b) dx = dt
∴ `I = (-1)/a int dt = (-1)/a t^2/2 + C`
`= (-cos^2 (ax + b))/(2a) + C`
APPEARS IN
RELATED QUESTIONS
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int 1/(sqrt"x" + "x")` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
`int sqrt(1 + "x"^2) "dx"` =
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int 1/(cos x - sin x)` dx = _______________
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int(5x + 2)/(3x - 4) dx` = ______
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 ______.
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate the following.
`intxsqrt(1+x^2)dx`
The value of `int ("d"x)/(sqrt(1 - x))` is ______.
Evaluate `int1/(x(x-1))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`
