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

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In summary, solving trigonometric integrals with sums allows us to find the area under a curve involving trigonometric functions, which has practical applications in fields such as physics and engineering. This can be done using techniques such as substitution, integration by parts, and trigonometric identities. The integral in the given equation can be solved analytically using integration by parts and can be solved for any real value of x except for 0. Solving trigonometric integrals with sums has various applications in real-world scenarios involving trigonometric functions.
  • #1
calculus1967
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A Trigonometric Integral

I'm trying to find
[tex]\int_3^x \frac{\sin t}{t}dt[/tex]
I can't find its indefinite integral:
So I'm trying to use
[tex]F(x)=\int_3^x\frac{\sin t}{t}dt[/tex]
and solve it using sums. I am wondering, will that will work?
 
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  • #2
It's of bad form to put the variable of integration inside the limit itself.

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

If

[tex]x = 0 \quad \text{or} \quad \Re(x) > 0 \quad \text{or} \quad \Im(x) \neq 0[/tex]

Not too exciting really.
 
  • #3


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.
 

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

1. What is the purpose of solving trigonometric integrals with sums?

The purpose of solving trigonometric integrals with sums is to find the area under a curve that involves trigonometric functions. This can be useful in real-world applications, such as calculating the work done by a varying force or the displacement of a vibrating object.

2. How do you solve a trigonometric integral with sums?

To solve a trigonometric integral with sums, you can use various techniques such as substitution, integration by parts, or trigonometric identities. The specific method used will depend on the form of the integral and the trigonometric function involved.

3. Can the integral in the given equation, F(x)=\int_3^x\frac{\sin t}{t}dt, be solved analytically?

Yes, the integral can be solved analytically using integration by parts. By letting u = \sin t and dv = \frac{dt}{t}, you can simplify the integral to get F(x) = \frac{\cos x}{x} - \frac{\cos 3}{3}.

4. Is there a specific range of values for x in which the given integral can be solved?

The given integral can be solved for any real value of x. However, if x = 0, the integral is undefined due to the division by 0 in the denominator.

5. What are the applications of solving trigonometric integrals with sums?

Solving trigonometric integrals with sums has various applications in physics, engineering, and other fields. It can be used to calculate quantities such as work, displacement, and velocity in real-world scenarios that involve trigonometric functions.

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