Stumped by Definite Integral: $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$

In summary, the definite integral in the given conversation is $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$. The speaker mentions trying a substitution using the inverse sin function, but it does not simplify the integral. The approximate numerical value is $0.9597$ and without technology, it is difficult to find a solution.
  • #1
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Was asked to solve this definite integral in a tech free test. Not sure how to go about it.

$$\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx.$$

I know here is a relationship between inverse sin and the sqrt function but with just sin x?
 
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  • #2
Well, Mathematica balks at evaluating it, so it seems to me it is a rather non-trivial integral. I tried the substitution $y=\arccsc(x)$, which changes the integral to
$$\int_{\pi/6}^{\pi/2}\sin(\csc(y)) \, dy.$$
Problem is, this new integral is no easier than the old one. The numerical value is about $0.9597$. If you're not allowed any technology, I don't see how you could arrive at any solution.
 

FAQ: Stumped by Definite Integral: $\int_1^2 \frac{\sin(x)}{\sqrt{x^2-1}} \, dx$

What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two specific points on the x-axis. It is denoted by ∫ and consists of a function, limits of integration, and a differential. It is used to calculate the total change in a function over a specific interval.

How do you solve a definite integral?

To solve a definite integral, you need to first find the antiderivative of the given function. Then, substitute the limits of integration into the antiderivative and calculate the resulting expression. This will give you the numerical value of the definite integral.

What is the function inside the integral?

The function inside the integral is f(x) = sin(x)/√(x^2-1). This is a trigonometric function with a radical expression in the denominator. It is a common type of function used in calculus and can be solved using various techniques such as substitution, integration by parts, and trigonometric identities.

Why is the definite integral important?

The definite integral has many important applications in mathematics, physics, and engineering. It is used to calculate the area under a curve, which is useful in finding the volume of irregular shapes and calculating work done in physics. It is also used to solve optimization problems and in the process of finding the average value of a function.

What is the significance of the limits of integration?

The limits of integration in a definite integral represent the interval over which the function is being integrated. They determine the starting and ending points of the integration and help in calculating the total change in the function over that specific interval. The choice of limits can greatly affect the numerical value of the definite integral.

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