Advertisements
Advertisements
Question
Find the integrals of the function:
sin x sin 2x sin 3x
Advertisements
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:
sin2 (2x + 5)
Find the integrals of the function:
sin4 x
Find the integrals of the function:
cos4 2x
Find the integrals of the function:
tan3 2x sec 2x
Find the integrals of the function:
tan4x
Find the integrals of the function:
`(sin^3 x + cos^3 x)/(sin^2x cos^2 x)`
Find the integrals of the function:
`(cos 2x+ 2sin^2x)/(cos^2 x)`
Find the integrals of the function:
`1/(sin xcos^3 x)`
Evaluate: `int_0^π (x sin x)/(1 + cos^2x) dx`.
Evaluate `int_0^(3/2) |x sin pix|dx`
Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx
Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot cosec x}dx\] .
Find `int_ (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `
Find `int_ sin ("x" - a)/(sin ("x" + a )) d"x"`
Find `int_ (log "x")^2 d"x"`
Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`
Find:
`int"dx"/sqrt(5-4"x" - 2"x"^2)`
Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`
Evaluate the following:
`int ("d"x)/(1 + cos 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`
`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.
`int sinx/(3 + 4cos^2x) "d"x` = ______.
The value of the integral `int_(1/3)^1 (x - x^3)^(1/3)/x^4 dx` is
What does integration using trigonometric identities mean?
Which identity is used for \(\cos^2 x\)?
Which identity is used for an even power of cosine?
Which identity is used for \(\sin^3 x\)?
What is \(\int \cos^2 x\,dx\)?
What is \(\int \sin 2x\cos 3x\,dx\)?
What should be done first when evaluating an integral containing a complicated trigonometric expression?
After a trigonometric expression has been simplified, how should it be integrated?
What must always be added to an indefinite integral?
