×

# Any one with a method

$\Large {\displaystyle \int^{b}_{0} \ln(\sin ax) \, dx} = \ ?$

Note by Tanishq Varshney
2 years, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

@Tanishq Varshney I don't think so this integration is possible with normal methods. Try substituting: $sinax=\frac{e^{iax}-e^{-iax}}{2i}$

- 2 years, 8 months ago

using complex numbers , its easy.

- 1 year ago

I can integrate that, if $$a = 1$$ or $$2$$, and $$b = \dfrac{n\pi}{2}$$. In that case, the answer to the integral would be $$= - \dfrac{n\pi}{2} \ln(2)$$.

I don't think I can integrate it for other values of $$a, b \neq 0$$.

- 2 years, 8 months ago

It is a bit surprising that this can be integrated at all, since the function ln(sin(x)) becomes infinite as x -> 0. (But this can be proven by considering the integral of ln(x) from 0 to b.)

Although I doubt that the function ln(sin(ax)) can be integrated explicitly in terms of "elementary" functions — and we better assume that a and b are both positive here ((or else both negative. But we will assume they are both positive).

Now assume a = 1. We want the function ln(sin(x)) to be real, and for this we need to avoid values of sin(x) that are less than 0. So we must limit the values of x near 0 to 0 <= x <= pi.

So let us set F(x) := the integral from 0 to x of ln(sin(t)) dt.

We know that F(0) = 0. By using the advanced complex variables technique of contour integration (in an advanced way), it can be determined that F(pi/2) = -pi ln(2)/2 and, by the symmetry of sin(x), that also F(pi) = -pi ln(2).

If we like, we can extend this function to all x via F(x) := the integral from 0 to x of ln|sin(t)| dt.

Then the graph of y = F(x) is a beautiful wavy curve that wiggles about the line y = -ln(2) x, and takes the exact values of F(N pi/2) = -N pi ln(2)/2 for all integers N.

Extending these facts to values of a other than a = 1 is an easy task.

- 2 years, 8 months ago

Let the integral be I. Now differentiate it with respect to a and after solving some more steps, you arrive at a differential equation in I and a i.e.,

dI/da + I/a = b*ln(sin(ab)).

Solve this differential equation to get I.

P.S.- Sorry, I dunno latex....else I would've posted my solution.

- 2 years, 8 months ago