Exact Solution for Integral of sin(x)/(10x+1) without Taylor Series?

[eric222]

Homework Statement

The problem: finding an exact result (or at least a method for computing the exact result) of

$$\int_0^5 \frac{sin(x)}{10x-1} dx$$

Homework Equations

Is there any way to solve this integral other than using Taylor series?

The Attempt at a Solution

$$= \int_0^5 \sum_{i=0}^{\infty} \frac{(-1)^ix^{2i+1}}{(2i+1)!(10x+1)}dx$$

 

[Char. Limit]

Homework Statement

The problem: finding an exact result (or at least a method for computing the exact result) of

$$\int_0^5 \frac{sin(x)}{10x-1} dx$$

Homework Equations

Is there any way to solve this integral other than using Taylor series?

The Attempt at a Solution

$$= \int_0^5 \sum_{i=0}^{\infty} \frac{(-1)^ix^{2i+1}}{(2i+1)!(10x+1)}dx$$

Hmm... let's try something...

$$\int_0^5 \frac{sin(x)}{10x-1} dx$$

u = x - 1/10

$$\int_{-.1}^{4.9} \frac{sin(u+.1)}{10u} dx$$

Whatever happens, though, it won't be an elementary integral, I'll tell you that right now. You can split the sin(u+.1) using the addition formula sin(x+y) = sin(x)cos(y) + cos(x)sin(y), but it won't be elementary.

 

[quZz]

Do you mean finding the principle value of this integral? What happens at x = 1/10...

 

[eric222]

Do you mean finding the principle value of this integral? What happens at x = 1/10...

How stupid of me. I should have noticed this directly.

Thanks for the help, everyone!