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}