Solve Trigonometric Integral with Sums: F(x)=\int_3^x\frac{\sin t}{t}dt

[calculus1967]

A Trigonometric Integral

I'm trying to find
\int_3^x \frac{\sin t}{t}dt
I can't find its indefinite integral:
So I'm trying to use
F(x)=\int_3^x\frac{\sin t}{t}dt
and solve it using sums. I am wondering, will that will work?

 

[Zurtex]

It's of bad form to put the variable of integration inside the limit itself.

\int_3^x \frac{\sin t}{t} dt = \text{SinIntegral}(x) - \text{SinIntegral}(3)

If

x = 0 \quad \text{or} \quad \Re(x) > 0 \quad \text{or} \quad \Im(x) \neq 0

Not too exciting really.

 

[M98Ranger]

Yes, using sums can work to solve this trigonometric integral. One approach is to use the Riemann sum method, where the interval [3, x] is divided into smaller subintervals and the function is approximated by the sum of the areas of rectangles under the curve. As the number of subintervals increases, the approximation becomes more accurate and approaches the actual value of the integral.

Another approach is to use the Taylor series expansion of sin t, which can be written as a sum of terms involving powers of t. This can then be integrated term by term to obtain an infinite series that converges to the value of the integral.

Overall, using sums to solve trigonometric integrals can be a useful technique, but it may not always be the most efficient or straightforward method. It is important to consider other techniques, such as integration by parts or trigonometric identities, to find the most efficient solution.