How do you find the antiderivative of #int sin^4xdx#?

Question

Evelyn Agard · Accepted Answer

#int sin^4xdx = - (sin^3x cosx )/4-3/8sinxcosx +3/8x+C#

As #d(cosx) = -sinx dx# we can write the integral as:

#int sin^4xdx = int sin^3 sinx dx = - int sin^3 d(cosx)#

and integrate by parts:

#int sin^4xdx = - sin^3x cosx + 3 int sin^2x cos^2x dx#

Now applying the identity:

#cos^2x = 1 - sin^2x#

#int sin^4xdx = - sin^3x cosx + 3 int sin^2x (1-sin^2x) dx#

and as the integral is linear:

#int sin^4xdx = - sin^3x cosx + 3 int sin^2xdx -3intsin^4x dx#

we have now the integral on both sides and we can solve for it:

#int sin^4xdx = - (sin^3x cosx )/4+ 3/4 int sin^2xdx #

We can now apply the same process for the integral:

# int sin^2xdx = -int sinx (dcosx) = -sinxcosx + int cos^2xdx = -sinxcosx + int (1-sin^2x)dx = -sinxcosx + int dx - int sin^2xdx#

and we get:

# int sin^2xdx = -(sinxcosx)/2 +x/2+C'#

Putting it together:

#int sin^4xdx = - (sin^3x cosx )/4-3/8sinxcosx +3/8x+C#

Note that you can write this result in an interesting form: first we use the identity:

#2sinx cosx = sin2x#

#- (sin^3x cosx )/4-3/8sinxcosx +3/8x = - (sin^2x sin2x)/8 -3/16sin2x +3/8x#

than we use:

#sin^2x = (1-cos2x)/2#

#- (sin^2x sin2x)/8 -3/16sin2x +3/8x = - ((1-cos2x) sin2x)/16 -3/16sin2x +3/8x = 1/32sin4x - 1/4sin2x +3/8x#

Aurora Alvarado · Answer

undefined To find the antiderivative of ∫sin^4(x) dx, you can use the power reduction formula for trigonometric functions. The formula states that sin^2(x) = (1 - cos(2x)) / 2. Applying this formula twice will help simplify the integral.

Here's the step-by-step process:

1. Use the power reduction formula for sin^2(x): sin^2(x) = (1 - cos(2x)) / 2.
2. Substitute sin^2(x) with (1 - cos(2x)) / 2 in the integral: ∫sin^4(x) dx = ∫(sin^2(x))^2 dx.
3. Substitute sin^2(x) with (1 - cos(2x)) / 2 again: ∫(sin^2(x))^2 dx = ∫((1 - cos(2x)) / 2)^2 dx.
4. Expand and simplify the expression: ∫((1 - cos(2x))^2 / 4) dx.
5. Expand and integrate: ∫(1 - 2cos(2x) + cos^2(2x)) / 4 dx.
6. Integrate each term separately: 
   - ∫(1/4) dx = (1/4)x.
   - ∫(-1/2 cos(2x)) dx = (-1/4)sin(2x).
   - ∫(1/4 cos^2(2x)) dx = (1/8)(2x + sin(4x)).
7. Combine the results: (1/4)x - (1/4)sin(2x) + (1/8)(2x + sin(4x)) + C.
   
Where C is the constant of integration.