what is the integral of of [(sin(x))^4+(cos(x))^4]/sin(x)+cos(x)?

Answer 1

sin^4(x)+cos^4(x)=1-2sin^2(x)cos^2(x)
now put sin^2x=(1-cos2x)/2 and cosx=(1+cos2x)/2
2sin^2xcos^2x=2(1/4)(1-cos^2x)=(1/2)[1-{(1+cos4x)/2}]
1/2-1/4(1+cos4x)
the integral=x/2+x/4+1/16cos4x+C
=3x/4+(cos4x/16)+C
this is the method
you can cross check my simplification

Answer 2

It may be easiest to go for complex exponentials and write

2 i sin x = [exp(ix) - exp(-ix)]
2 cos x = [exp(ix) + exp(-ix)]

so

16 (sin x)^4 = [exp(4ix) - 4exp(2ix) + 6 - 4exp(-2ix) + exp(-4ix)]

and 16 (cos x)^4 similar with pluses only. Adding the two, we have

16 [(sin x)^4 + (cos x)^4] = 2 exp(4ix) + 12 + 2 exp(-4ix)

which is equal to

12 + 4 cos (4x)

The integral is

12 x + sin (4x) + C.

Now divide by 16,

3/4 + 1/16 sin (4x) + C.


THe other solution given is almost correct, except that the integral of cosine is sine, not cosine :)

Answer 3

Let t=sin(x) then
(cos(x))^4 = ((cos(x))^2)^2 = (1-t^2)^2
Multiply and simplify to get
2(sin(x)^3) - 2(sin(x))^2 + (sin(x))^-2
Break it up into the three integrals and run with it.


Doug

Answer 4

42

Answer 5

So disappointed no one has answered this for you yet.
I so wanted to know the answer

Answer 6

hey doug you are wrong what happens to dt

Answer 7

That's tough one!!!!!! Will crack it once you have anwers