what is the integral of sin x over x dx?

Answer 1

Integral [ (sin(x)/x) dx ]

The fact of the matter is, this integral is one that cannot be expressed in terms of elementary functions. There's no way we can solve this using the methods we know; we cannot use integration by parts, partial fractions, substitution, trigonometric substitution, etc to solve this.

We can, however, approximate the integral through a power series. sin(x) has its own power series, so all we need to do is divide each term of the series by x (this represents (1/x)sin(x), or sin(x)/x) and then integrate thereafter.

como: Your proposed solution doesn't work, and here's why. Let

f(x) = (1/x)cos x - (1/x²)sinx + C

To make it easier to differentiate, factor (1/x).

f(x) = (1/x) [cos(x) - (1/x)sin(x)] + C

Differentiate using the product rule, noting that d/dx (1/x) = -1/x^2 gives us

f'(x) = (-1/x^2) [cos(x) - (1/x)sin(x)] + (1/x) [-sin(x) - [(-1/x^2)sin(x) + (1/x)cos(x)] ]

f'(x) = -cos(x)/x^2 + sin(x)/x^3 - sin(x)/x + sin(x)/x^2 - cos(x)/x

And as you can see, it looks nothing like sin(x)/x.

Answer 2

Integral Sinx X

Answer 3

For f(x) = (sin x)/x
f(x) is known as the sinc function, sadly the antiderivative of this function cannot be evaluated with elementary analytic methods.
And the quotient rule can only be used for differentiation.

You may try contour integration or quite simply find the MacLaurin expansion for sin x, divide throughout by x, and integrate the resulting series term by term. And there's your answer!

Answer 4

I = ∫ sinx / x dx --------(use quotient rule)

I = (x.cosx - sinx.1) / x² + C

I = (1/x).cos x - (1/x²).sinx + C