Verified answer

∫ 3 sin^4(x) cos^2(x) dx =

∫ 3 sin^4(x) (1-sin^2(x) ) dx =

3 ∫ (sin^4(x) - sin^6(x) ) dx =

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Now we apply the reduction formula for the sin^n (x). Which is:

[∫ sin^n(x)= (-1/n) sin^(n-1) (x) cos(x)+ (n-1)/n ∫ sin^(n-2)(x) dx ]

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3 ∫ (sin^4(x) - sin^6(x) ) dx =

3 [ ∫ sin^4(x) dx - ∫ sin^6(x) dx ] =

Now, we can apply the formula to both. But notice the formula reduces the power by 2 each time. So we can choose to evaluate the 2nd integral (involving sine to the 6th power) and combine alike terms.

∫ sin^6(x) dx =(-1/6)sin^5(x)cos(x)+5/6 ∫ sin^4x dx

Therefore,

3 [ ∫ sin^4(x) dx - ∫ sin^6(x) dx ] =

3 [ ∫ sin^4(x) dx +(1/6)sin^5(x)cos(x)- 5/6 ∫ sin^4x dx] =

3 [(1/6) ∫ sin^4(x) dx +(1/6)sin^5(x)cos(x) ] = , ( we added the integrals involving sin^4x )

Now applying the reduction again, for sin^4x we have:

(1/2) ∫ sin^4(x) dx +(1/2)sin^5(x)cos(x) =

(-1/8) sin^3(x)cos(x)+ 3/4 ∫ sin^2(x)dx + (1/2)sin^5(x)cos(x) =

Now applying the reduction again, for sin^2x we have:

∫ sin^2(x) dx = (-1/2)sin(x)cos(x)+(1/2)∫dx =

(-1/2)sin(x)cos(x)+(1/2) x.

Now plugging this back into the previous equation, we have:

(-1/8) sin^3(x)cos(x)+ 3/4 ((-1/2)sin(x)cos(x)+(1/2) x)+ (1/2)sin^5(x)cos(x) + C=

-sin^3(x)cos(x)/8 -3sin(x)cos(x)/8 + (3/4)x+(1/2)sin^5(x)cos(x) + C [ANSWER]

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Remarks:

(i) Because of trigonometric identities there is many ways to display the answer. All of which are equal.

(ii) The reduction formula can be proved by integration by parts.

3 * sin(x)^4 * cos(x)^2 * dx =>

3 * sin(2x/2)^4 * cos(2x/2)^2 * dx =>

3 * (1/2)^2 * (1 - cos(2x))^2 * (1/2) * (1 + cos(2x)) * dx =>

3 * (1/8) * (1 - cos(2x)^2) * (1 - cos(2x)) * dx =>

(3/8) * (sin(2x)^2) * (1 - cos(2x)) * dx =>

(3/8) * sin(2x)^2 * dx - (3/8) * sin(2x)^2 * cos(2x) * dx =>

(3/8) * sin(4x/2)^2 * dx - (3/8) * sin(2x)^2 * cos(2x) * dx =>

(3/8) * (1/2) * (1 - cos(4x)) * dx - (3/8) * sin(2x)^2 * cos(2x) * dx =>

(3/16) * dx - (3/16) * cos(4x) * dx - (3/8) * sin(2x)^2 * cos(2x) * dx

u = sin(2x)

du = 2 * cos(2x) * dx

(3/16) * dx - (3/16) * cos(4x) * dx - (3/8) * (1/2) * u^2 * du =>

(3/16) * (dx - cos(4x) - u^2 * du)

Integrate

(3/16) * (x - (1/4) * sin(4x) - (1/3) * u^3) + C =>

(3/16) * (1/4) * (1/3) * (12 * x - 3 * sin(4x) - 4 * u^3) + C =>

(3/192) * (12x - 3sin(4x) - 4sin(2x)^3) + C