Easy Guide: Solving Integral (sin x)/x dx from 0 to pi | Tips & Tricks

[lueffy]

How to solve the integral (sin x) / x dx
from 0 to pi (3.14...)

at first i thought this could be easily done with partial integration,
but on the contrary, I'm still dying here, trying to solve it...

 

[VietDao29]

Nope. It cannot be done by Integration by parts (note that it's Integration by parts and not Partial integration). One way to do it is to notice that:
$$\sin x = x - \frac{x ^ 3}{3!} + \frac{x ^ 5}{5!} + ... = \sum_{k = 0} ^ {\infty} (-1) ^ {k} \frac{x ^ {2k + 1}}{(2k + 1)!}$$
So:
$$\frac{\sin x}{x} = 1 - \frac{x ^ 2}{3!} + \frac{x ^ 4}{5!} + ... = \sum_{k = 0} ^ {\infty} (-1) ^ {k} \frac{x ^ {2k}}{(2k + 1)!}$$
Now, just integrate both sides, can you go from here? :)
$$\int \frac{\sin x}{x} dx = \int \left( \sum_{k = 0} ^ {\infty} (-1) ^ {k} \frac{x ^ {2k}}{(2k + 1)!} \right) dx = ?$$

 

[lueffy]

Wow, thanks man, but that's just beyond my calculus class...
But, whew, it really wasn't an easy one...
Thanks for telling me about the "partial integration" part...

But how you suppose to do the latter integration?
an integral with sigma inside? i thought that integral sign is somewhat related to sigma, but only for continuous distribution of the partition, from the definition of Riemann's integrals...

 

[lueffy]

Eh, and may i ask you one more thing?
Why the integral can't be done with integration by parts?
i mean, this far i just know few methods to solve integrals, that is substituting, integration by parts, and with trig subs...
I heard there's a lot of integral types, which can't be done with only the very few methods that I've mentioned above...

Thanks for the help, I'm greatly appreciated it...

 

[Muzza]

(note that it's Integration by parts and not Partial integration).

Nonsense...

 

[VietDao29]

Wow, thanks man, but that's just beyond my calculus class...
But, whew, it really wasn't an easy one...
Thanks for telling me about the "partial integration" part...

But how you suppose to do the latter integration?
an integral with sigma inside? i thought that integral sign is somewhat related to sigma, but only for continuous distribution of the partition, from the definition of Riemann's integrals...

This is correct. Just do it like you are integrating x dx, or (x² + 3x) dx, ...
$$\int \frac{\sin x}{x} dx = \sum_{k = 0} ^ \infty \left( (-1) ^ k \frac{x ^ {2k + 1}}{(2k + 1) (2k + 1)!} \right) + C$$

Why the integral can't be done with integration by parts?
i mean, this far i just know few methods to solve integrals, that is substituting, integration by parts, and with trig subs...

It's because there's no elementary function, whose derivative is sin(x) / x.
One can also define a Sine Integral to be:
$$\mbox{Si} (x) = \int \limits_{0} ^ x \frac{\sin t}{t} dt$$

Nonsense...

Nah, I've so far heard of partial fraction, partial derivative, but not partial integration. It's just ill-worded, and not formal. Yes, one can choose to say partial integration, or integrate by parts, it happens that both are okay. But I myself prefer integrate by parts. And hopefully, most professors and/or mathematicians agree with me!
By the way, is guiding a guy to word formally worth being called nonsense?

 

[Muzza]

Nah, I've so far heard of partial fraction, partial derivative, but not partial integration.

A phrase can be in use even if you've never heard of it. Mathworld recognizes it as a synonym of "integration by parts", for example.

It's just ill-worded, and not formal.

You've got to be kidding me...