Question

Find \[\int_0^\pi {\left| {\cos x} \right|} \,dx = \]
A. \[\dfrac{1}{2}\]
B. \[ - 2\]
C. \[1\]
D. \[ - 1\]
E. \[2\]

Answer 1

Hint: We have been given to calculate the definite integral of a trigonometric function, which is a cosine function. As we see that the given function is a modulus function, we first break the integral into two parts, then integrate the given function, and finally apply upper and lower limits of integration.

Formula used:
We have been using the following formulas:
1. \[\int {\cos x = \sin x + c} \]

Complete step-by-step solution:
We are given that \[\int_0^\pi {\left| {\cos x} \right|} \,dx\]
Now we divide the integral into two parts, and we get
\[\int_0^\pi {\left| {\cos x} \right|} \,dx = \int_0^{\dfrac{\pi }{2}} {\left| {\cos x} \right|} \,dx + \int_{\dfrac{\pi }{2}}^\pi {\left| {\cos x} \right|} \,dx\]
Now we know that the function
 \[\left|cos x \right| = cos x\] if \[ 0 < x < \pi/2\]
\[\left|cos x \right| = - cos x\] if \[\pi/2 < x < \pi \]
Therefore, we get
\[\int_0^\pi {\left| {\cos x} \right|} \,dx = \int\limits_0^{\dfrac{\pi }{2}} {\cos x\,dx - } \int\limits_{\dfrac{\pi }{2}}^\pi {\cos x\,dx...\left( 1 \right)} \]
Now we know that \[\int {\cos x = \sin x} \]
Therefore, by integrating equation (1), we get
\[
  \int_0^\pi {\left| {\cos x} \right|} \,dx = \left[ {\sin x} \right]_0^{\dfrac{\pi }{2}} - \left[ {\sin x} \right]_{\dfrac{\pi }{2}}^\pi \\
   = \left[ {\sin \dfrac{\pi }{2} - \sin \,0} \right] - \left[ {\sin \pi - \sin \dfrac{\pi }{2}} \right] \\
 \]
Now we know that
\[
  \sin \dfrac{\pi }{2} = 1 \\
  \sin \pi = 0 \\
  \sin 0 = 0
 \]
Now by substituting these values and simplifying it, we get
\[
  \int_0^\pi {\left| {\cos x} \right|} \,dx = \left[ {1 - 0} \right] - \left[ {0 - 1} \right] \\
   = 1 - 0 - 0 + 1 \\
   = 1 + 1 \\
   = 2
 \]
Therefore, the value of \[\int_0^\pi {\left| {\cos x} \right|} \,dx\] is equal to \[2\]
Hence, option (E) is correct option

Additional information: The area bound under the function graph \[y = f\left( x \right)\], and above the x-axis, which is bound between two bounds as \[x = a\] and \[x = b\], is the definite integral of a function \[f\left( x \right)\]. In this case, \[a = 0\] and \[b = \dfrac{\pi }{2}\]. One of the capabilities of integral calculus is that when the function is integrated with respect to \[dx\], one of the two boundaries of the area to be found is a curve and the other is the x-axis.

Note: In such questions, the key concept is to always remember all of the properties of definite integrals then simplify the integral, and also remember the integration formula of trigonometric function and values then simplify for the required result.