Advertisements
Advertisements
Question
Find the integrals of the function:
sin x sin 2x sin 3x
Solution
Let `I = int sin x sin 2x sin 3x dx`
`= 1/2 int (2 sin x sin 2x) sin 3x dx`
`= 1/2 int (cos x - cos 3x) sin 3x dx` ... [∵ 2 sin A sin B = cos (A - B) - cos (A + B)]
`= 1/4 int 2 sin 3x cos x dx - 1/4 int 2 sin 3x cos 3x dx` .... [∵ 2 sin A cos B = sin (A + B) + sin (A - B)]
`= 1/4 int (sin 4x + sin 2x) dx - 1/4 int sin 6x dx`
`= -1/16 cos 4x - 1/8 cos 2x + 1/24 cos 6x + C`
`= 1/4 [1/6 cos 6x - 1/4 cos 4x - 1/2 cos 2x] + C`
APPEARS IN
RELATED QUESTIONS
Evaluate : `intsin(x-a)/sin(x+a)dx`
Find the integrals of the function:
sin3 (2x + 1)
Find the integrals of the function:
sin3 x cos3 x
Find the integrals of the function:
sin 4x sin 8x
Find the integrals of the function:
`(1-cosx)/(1 + cos x)`
Find the integrals of the function:
`cos x/(1 + cos x)`
Find the integrals of the function:
`(sin^2 x)/(1 + cos x)`
Find the integrals of the function:
`(cos 2x - cos 2 alpha)/(cos x - cos alpha)`
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 x)`
Find the integrals of the function:
sin−1 (cos x)
`int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx` is equal to ______.
`int (e^x(1 +x))/cos^2(e^x x) dx` equals ______.
Find `int dx/(x^2 + 4x + 8)`
Evaluate `int_0^pi (x sin x)/(1 + cos^2 x) dx`
Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx
Find `int_ sin ("x" - a)/(sin ("x" + a )) d"x"`
Find `int_ (log "x")^2 d"x"`
Find `int_ (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`
Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Find:
`int"dx"/sqrt(5-4"x" - 2"x"^2)`
Evaluate `int tan^8 x sec^4 x"d"x`
`int "dx"/(sin^2x cos^2x)` is equal to ______.
`int (sin^6x)/(cos^8x) "d"x` = ______.
Evaluate the following:
`int ((1 + cosx))/(x + sinx) "d"x`
Evaluate the following:
`int ("d"x)/(1 + cos x)`
Evaluate the following:
`int tan^2x sec^4 x"d"x`
Evaluate the following:
`int (sinx + cosx)/sqrt(1 + sin 2x) "d"x`
Evaluate the following:
`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`
Evaluate the following:
`int (cosx - cos2x)/(1 - cosx) "d"x`
Evaluate the following:
`int sin^-1 sqrt(x/("a" + x)) "d"x` (Hint: Put x = a tan2θ)
`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.
