The Definite Integral of Cosx from 0 to Pi

In this tutorial we shall derive the definite integral of the trigonometric function cosine from limits 0 to Pi.

The integration of the form is
\[I = \int\limits_0^\pi {\cos xdx} \]

First we evaluate this integration by using the integral formula $$\int {\cos xdx = \sin x} $$, and then we use the basic rule of the definite integral $$\int\limits_a^b {f\left( x \right)dx = \left| {F\left( x \right)} \right|_a^b} = \left[ {F\left( b \right) – F\left( a \right)} \right]$$. So we have
\[\begin{gathered} \int\limits_0^\pi {\cos xdx} = \left| {\sin x} \right|_0^\pi \\ \Rightarrow \int\limits_0^\pi {\cos xdx} = \left[ {\sin \pi – \sin 0} \right] \\ \end{gathered} \]

From the trigonometric values \[\sin \pi = 0\] and $$\sin 0 = 0$$, we have
\[\begin{gathered} \Rightarrow \int\limits_0^\pi {\cos xdx} = \left[ {0 – 0} \right] \\ \Rightarrow \int\limits_0^\pi {\cos xdx} = \left[ 0 \right] \\ \Rightarrow \int\limits_0^\pi {\cos xdx} = 0 \\ \end{gathered} \]