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vee6
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How to find the integral of sin x dx?
$$\int sin x dx=?$$
$$\int sin x dx=?$$
vee6 said:How to find the integral of sin x dx?
$$\int sin x dx=?$$
Do you know a function whose derivative is sin(x)? That would be the antiderivative you want.vee6 said:How to find the integral of sin x dx?
$$\int sin x dx=?$$
To find it? There are two ways I can think of:vee6 said:How to find the integral of sin x dx?
$$\int sin x dx=?$$
Mandelbroth said:To find it? There are two ways I can think of:
1) Recognize that ##\frac{d}{dx}[-\cos(x)+C]=\sin(x)##
2) See that ##\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}##, and then integrate the complex exponentials.
Mandelbroth said:2) See that ##\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}##, and then integrate the complex exponentials.
lukka said:The integral of e^ix is i^3(e^ix) +c = -i(e^ix) +c
lukka said:since i^3 (i) = i^4 = 1
dy/dx [i^3(e^ix)] = e^ix
lukka said:I'm not sure you can for anything but the most vigorous proof Micro, but one will often know from experience that the exponential function that which is a weighted sum of the sine and cosine, equals it's own derivative. Maybe someone else could provide an answer to this..
I'd use a Taylor series. As usual, micro's point is valid. However, if we assume the exponential function is holomorphic in the whole complex plane (I'm presenting this without much rigorous proof because I'm feeling a little lazy right now), it follows that the derivative of the exponential function in the complex plane is the same as the derivative of its Taylor series, id estlukka said:Yes i understand what you are saying. That's a valid point you have made. So how would one go about that Micro?
Mandelbroth said:I'd use a Taylor series. As usual, micro's point is valid. However, if we assume the exponential function is holomorphic in the whole complex plane (I'm presenting this without much rigorous proof because I'm feeling a little lazy right now), it follows that the derivative of the exponential function in the complex plane is the same as the derivative of its Taylor series, id est
$$\frac{d(e^{ix})}{dx}=\frac{d}{dx}\left[\sum_{n=0}^{\infty}\frac{(ix)^n}{n!}\right]=\sum_{n=0}^{\infty}\frac{i^{n+1}x^n}{n!}=ie^{ix}.$$
Again, no. I'm assuming they know the integral of a function of the form ##f(x)=e^{ax}##. Do you think someone who would ask such a thing would be doing a rigorous proof of the concept? I was providing my method of doing it, because I think it works.micromass said:Right. And then you need to know Taylor series. Additionally, you must be able to justify why you can exchange derivative and series. Do you think that somebody who just asked what the derivative of sin(x) is, knows these things?
Mandelbroth said:Again, no. I'm assuming they know the integral of a function of the form ##f(x)=e^{ax}##. Do you think someone who would ask such a thing would be doing a rigorous proof of the concept? I was providing my method of doing it, because I think it works.
I'm letting him or her assume, without proof, that integration works similarly for complex ##a##. Again, the idea of the proof was for you, because you were asking how one finds the derivative of ##e^{ix}## without sines and cosines. I was also attempting to convey that, at this point, I don't think the OP is looking for a rigorous proof.micromass said:People in calculus typically only see this for real numbers ##a## and ##x##. Not for complex variables.
I know your proof works. I'm just saying it's not helpful to the OP.
Mandelbroth said:Now, let's ... actually help the OP. Alright?
micromass said:I know your proof works. I'm just saying it's not helpful to the OP.
To find the integral of sin x dx, you can use the integration by parts method or the substitution method. Both methods involve following a set of steps and applying the appropriate rules to solve the integral.
The steps for finding the integral of sin x dx are:
1. Identify the integral and determine which method to use
2. Apply the appropriate integration rule
3. Solve for the integral using algebraic manipulations
4. Check your solution by differentiating to see if it matches the original function.
Yes, you can use the trigonometric identity sin^2 x + cos^2 x = 1 to simplify the integral of sin x dx. This can make the integration process easier and faster.
The general formula for finding the integral of sin x dx is:
∫ sin x dx = - cos x + C, where C is the constant of integration.
Yes, it is possible to solve the integral of sin x dx without using any trigonometric identities. This can be done by using the substitution method, where you substitute sin x with u, and then solve for the integral using algebraic manipulations.