# The Definite Integral of Sinx from 0 to Pi

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

The integration of the form is
$I = \int\limits_0^\pi {\sin xdx}$

First we evaluate this integration by using the integral formula $\int {\sin xdx = – \cos x}$, 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
$\begin{gathered} \int\limits_0^\pi {\sin xdx} = \left| { – \cos x} \right|_0^\pi \\ \Rightarrow \int\limits_0^\pi {\sin xdx} = – \left| {\cos x} \right|_0^\pi \\ \Rightarrow \int\limits_0^\pi {\sin xdx} = – \left[ {\cos \pi – \cos 0} \right] \\ \end{gathered}$

From the trigonometric values $\cos \pi = – 1$ and $\cos 0 = 1$, we have
$\begin{gathered} \Rightarrow \int\limits_0^\pi {\sin xdx} = – \left[ { – 1 – 1} \right] \\ \Rightarrow \int\limits_0^\pi {\sin xdx} = – \left[ { – 2} \right] \\ \Rightarrow \int\limits_0^\pi {\sin xdx} = 2 \\ \end{gathered}$