Problem

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x \sin x}{1+x^{8}} d x$

Solution

Step 1 :Define the function $f(x) = \frac{x \sin x}{1+x^{8}}$ and the limits of integration $a = -\frac{\pi}{4}$ and $b = \frac{\pi}{4}$

Step 2 :Choose a value for $n$ (the number of subintervals), for example, $n = 1000$

Step 3 :Calculate the step size $h = \frac{b-a}{n} = \frac{\frac{\pi}{4} - (-\frac{\pi}{4})}{1000} = 0.0015707963267948967$

Step 4 :Calculate the $x$ values for each subinterval: $x_i = a + ih$ for $i = 0, 1, \dots, n$

Step 5 :Use the trapezoidal rule to approximate the integral: $\int_{a}^{b} f(x) dx \approx \frac{h}{2} \sum_{i=1}^{n} (f(x_{i-1}) + f(x_i))$

Step 6 :Calculate the approximate value of the integral: $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x \sin x}{1+x^{8}} d x \approx 0.29270320329010313$

Step 7 :\(\boxed{0.2927}\)

From Solvely APP
Source: https://solvelyapp.com/problems/8378/

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