\n\\int_{1}^{5} \\frac{\\cos(x)}{x} dx\n
\n\\int_{1}^{5} \\frac{\\cos(x)}{x} dx\n
Answer
Explanation:
Step1: Recall the integral property
The integral $\int_{a}^{b}f(x)dx$ represents the net area under the curve $y = f(x)$ from $x=a$ to $x = b$. For the function $y=\frac{\cos(x)}{x}$, there is no elementary antiderivative. We can use numerical methods (such as the Trapezoidal rule or Simpson's rule) or refer to the sine - integral function. The sine - integral function is defined as $\text{Si}(x)=\int_{0}^{x}\frac{\sin(t)}{t}dt$. Another way is to use a calculator or software with integral - computing capabilities.
Using a calculator (for example, a TI - 89 or an online integral calculator like Wolfram Alpha) to evaluate the definite integral $\int_{1}^{5}\frac{\cos(x)}{x}dx$.
Answer:
The value of $\int_{1}^{5}\frac{\cos(x)}{x}dx\approx0.050$