Definite Integral of Sinx from 0 to Pi

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

The integration of the form

I =  \int\limits_0^\pi  {\sin xdx}


First we evaluate this integration by using integral formula \int {\sin xdx =  - \cos x} , then we using the basic rule of 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], 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 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}