Question

Evaluate the integral

\(\int\limits_{\rm{0}}^{\rm{\pi }} {{\rm{(5}}{{\rm{e}}^{\rm{x}}}{\rm{ + 3sinx)dx}}} \)

Short answer

The value of the integral is\({\rm{5}}{{\rm{e}}^{\rm{x}}}{\rm{ + 1}}\)

Step-by-step solution

Finding General antiderivative

Tofindthe general antiderivative using the indefinite integrals

\(\)\(\begin{array}{l}\int {{{\rm{e}}^{\rm{x}}}{\rm{dx = }}{{\rm{e}}^{\rm{x}}}{\rm{ + C}}} \\\int {{\rm{sinxdx = - cosx + C}}} \end{array}\)

and ignoring (just for the moment) the boundaries of integration.

Now

\(\begin{array}{c}\int {{\rm{(5}}{{\rm{e}}^{\rm{x}}}{\rm{ + 3sinx)dx}}} {\rm{ = 5}}\int {{{\rm{e}}^{\rm{x}}}{\rm{dx + 3}}\int {{\rm{sinxdx}}} } \\{\rm{ = 5}}{{\rm{e}}^{\rm{x}}}{\rm{ - 3cosx + C}}\\{\rm{ = F(x)}}\end{array}\)

We will leave out the arbitrary constant C, since it cancels out when calculating a definite integral.

Assigning final value.

Therefore from the evaluation theorem,

\(\begin{array}{c}\int\limits_{\rm{0}}^{\rm{\pi }} {{\rm{(5}}{{\rm{e}}^{\rm{x}}}{\rm{ + 3sinx)dx = }}\left( {{\rm{5}}{{\rm{e}}^{\rm{x}}}{\rm{ - 3cosx}}} \right)} _{\rm{0}}^{\rm{\pi }}{\rm{ = (5}}{{\rm{e}}^{\rm{\pi }}}{\rm{ - 3cos\pi ) - (5}}{{\rm{e}}^{\rm{0}}}{\rm{ - 3cos0)}}\\{\rm{ = (5}}{{\rm{e}}^{\rm{\pi }}}{\rm{ - 3( - 1)) - (5(1) - 3(1))}}\\{\rm{ = (5}}{{\rm{e}}^{\rm{\pi }}}{\rm{ + 3) - (5 - 3)}}\\{\rm{ = 5}}{{\rm{e}}^{\rm{\pi }}}{\rm{ + }}{{\rm{1}}^{}}\end{array}\)