Basic Integral without Basic Concept

A friend proposed this problem to me.

Find $\displaystyle\int \cos x \ \mathrm{d}x$ without using the fact that the derivative of sine is cosine.

cepheid
Staff Emeritus
Gold Member
A friend proposed this problem to me.

Find $\displaystyle\int \cos x \ \mathrm{d}x$ without using the fact that the derivative of sine is cosine.
How about using the Taylor series expansion of the cosine function?$$\int \cos x\,dx = \int \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\,dx$$$$= \sum_{n=0}^{\infty} \int \frac{(-1)^n x^{2n}}{(2n)!}\,dx$$$$= \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\frac{1}{2n+1}x^{2n+1}$$Now, (2n+1)*(2n)! = (2n+1)! so that this becomes$$\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$Now all you have to do is "notice" that this result is the Taylor series expansion of a sine function.

I just realized that you'd probably have to know the derivative of cosine in order to derive its Taylor series expansion in the first place, so I'm not sure whether you are satisfied with this answer.

A friend proposed this problem to me.

Find $\displaystyle\int \cos x \ \mathrm{d}x$ without using the fact that the derivative of sine is cosine.
I think this is not possible. Maybe your friend considers

$$\sin\theta=\frac{{e}^{i\theta}-{e}^{-i\theta}}{2i}$$
$$\cos\theta=\frac{{e}^{i\theta}+{e}^{-i\theta}}{2}$$

to be definitions of sine and cosine? But sine and cosine are defined to y/r and x/r respectively.

cepheid
Staff Emeritus
Gold Member
I think this is not possible. Maybe your friend considers

$$\sin\theta=\frac{{e}^{i\theta}-{e}^{-i\theta}}{2i}$$
$$\cos\theta=\frac{{e}^{i\theta}+{e}^{-i\theta}}{2}$$

to be definitions of sine and cosine?
Assuming we're allowed that definition of cosine, then that also works.

But sine and cosine are defined to y/r and x/r respectively.
I'm not sure I understand your statement here. Can you clarify?

HallsofIvy
Homework Helper
I think this is not possible. Maybe your friend considers

$$\sin\theta=\frac{{e}^{i\theta}-{e}^{-i\theta}}{2i}$$
$$\cos\theta=\frac{{e}^{i\theta}+{e}^{-i\theta}}{2}$$

to be definitions of sine and cosine? But sine and cosine are defined to y/r and x/r respectively.
Non-sense- because you haven't said what x, y, and r are. There are, in fact, many different ways to define sine and cosine. The formulas given above are perfectly good as definitions.

You could also define cos(x) as $\sum_{n=0}^\infty [(-1)^n/(2n)!]x^{2n}$
and sine(x) as $\sum_{n=0}^\infty [(-1)^n/(2n+1)!]x^{2n+1}$

Or you can define cos(x) as the function satifying y''= -y with initiial conditions y(0)= 1, y'(0)= 0 and define sin(x) as the function satisfying y''= y with initial conditions y(0)= 0, y'(0)= 1.

I'm not sure I understand your statement here. Can you clarify?
You haven't said what x, y, and r are.
If you draw a right-angled triangle, then x is the adjacent side, y is the opposite side and r is the hypotenuse.

Non-sense- because you haven't said what x, y, and r are. There are, in fact, many different ways to define sine and cosine. The formulas given above are perfectly good as definitions.

You could also define cos(x) as $\sum_{n=0}^\infty [(-1)^n/(2n)!]x^{2n}$
and sine(x) as $\sum_{n=0}^\infty [(-1)^n/(2n+1)!]x^{2n+1}$
Yes but I am considering the geometric definition.

define sin(x) as the function satisfying y''= y with initial conditions y(0)= 0, y'(0)= 1.
Isn't the second derivative of sin(x) is -sin(x) ?

Mark44
Mentor
Isn't the second derivative of sin(x) is -sin(x) ?
Yes. I'm pretty sure HallsOfIvy meant y'' + y = 0 as the differential equation.