Integrate (sin(x))^2?

Question

How do you integrate (sin(x))^2 WITHOUT trig identities?

Answer 1

Use the Euler formula.

Sin(x) = (1/2i)[exp(ix) - exp(-ix)]

Cos(x) = (1/2)[exp(ix) + exp(-ix)]

and exp(ix) = cos(x) + i*sin(x)

i is equal to the square root of -1. Treat it like any other constant when you integrate.

int [sin(x)]^2 = int[{(1/2i)^2}{exp(ix) - exp(-ix)}^2]

=(-1/4)int[exp(2ix) -2exp(0) + exp(-2ix)]

=(-1/4)int[exp(2ix)] + (-1/4)int[-2] + (-1/4)int[exp(-2ix)

=(-1/4)(1/2i)exp(2ix) + (-1/4)(-2x) + (-1/4)(-1/2i)exp(-2ix)

=(-1/8i)exp(2ix) + x/2 + (1/8i)exp(-2ix)

=x/2 + (1/8i)[-exp(2ix) + exp(-2ix)

=x/2 + (-1/4){(1/2i)[exp(2ix) - exp(-2ix)]}

= x/2 - (1/4)sin(2x)

Answer 2

write the power series for sin(x):

sin(x) = ∑(-1)^n x^(2n +1)/(2n+1)!

square it, and integrate with respect to x (all of the n's and the sum are constant ) so you have to integrate: x^2(2n +1)

when you are done you have to work a bit with the new sum that you'll get, but if you do you, you'll find the trig identities.

Answer 3

Use the following formula:

integral of sin^2 x dx = 1/2(x - sin x cos x) +C

G

Answer 4

The simple fact of the matter is that you can't. Trig identities are required in solving this problem, because you can't intergrate sin^2(x) directly.

Wanna change it to -cos^2(x)/2 using the power rule? Sorry; you can't, because the derivatiive of (-1/2) cos^2(x) is (-1/2)(2)cos(x)(-sin(x) = sin(x)cos(x).

You absolutely *have* to use the half angle identity.

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

Integral ( (1/2) (1 - cos2x))dx =
(1/2) * Integral (1 - cos2x)dx
(1/2) * [x - (1/2)sin(2x)] + C

The reason why we have to use trig identities is because sin^2(x) isn't on our list of known derivatives.

d/dx sin(x) = cos(x)
d/dx cos(x) = -sin(x)
d/dx tan(x) = sec^2(x)
d/dx cot(x) = -csc^2(x)
d/dx sec(x) = sec(x)tan(x)
d/dx csc(x) = -csc(x)cot(x)

As you can see, everything to the right of the equal sign has no sin^2(x). No amount of substitution, partial fractions, trig substitution, reverse power rule, and so forth, is going to make solving this integral any easier.

Answer 5

as quickly as I ask the different 2 I artwork with for his or her terrific, they answer, 'Huh?' it particularly is a lot down south, my buddy. the main i will wish for is an smart nod of somebody's head---no speech.

Answer 6

cosx*[(sinx)^3]/3