To evaluate the definite integral
[tex]\[ \int_{\pi / 4}^\pi \frac{\sin x}{1+\cos^2 x} \, dx \][/tex]
we can follow these steps:
1. Understand the Integrand:
The integrand is [tex]\(\frac{\sin x}{1 + \cos^2 x}\)[/tex]. We need to integrate this function with respect to [tex]\(x\)[/tex] from the lower limit [tex]\(\frac{\pi}{4}\)[/tex] to the upper limit [tex]\(\pi\)[/tex].
2. Set Up the Integral:
The integral is set up as:
[tex]\[
I = \int_{\pi / 4}^\pi \frac{\sin x}{1+\cos^2 x} \, dx
\][/tex]
3. Identifying an Appropriate Method:
To solve this integral, one typically needs to use substitution or special techniques, however in this context we will assume numerical integration methods are used, due to the complexity of the integrand.
4. Numerical Integration:
With the given bounds of integration [tex]\(\frac{\pi}{4}\)[/tex] and [tex]\(\pi\)[/tex], numerical integration techniques such as Simpson's rule, Trapezoidal rule, or methods implemented in computational tools can be used to evaluate the integral.
5. Compute the Result:
After performing the numerical integration, the result obtained for the integral [tex]\(\int_{\pi / 4}^\pi \frac{\sin x}{1+\cos^2 x} \, dx\)[/tex] is approximately [tex]\(1.4008778720678356\)[/tex].
6. Rounding the Result:
Finally, we round this result to three decimal places to get:
[tex]\[
\boxed{1.401}
\][/tex]
Thus, the value of the definite integral [tex]\(\int_{\pi / 4}^\pi \frac{\sin x}{1+\cos^2 x} \, dx\)[/tex], rounded to three decimal places, is [tex]\(1.401\)[/tex].
