Mathematics

Evaluate the following definite integrals as limit of sums.
$$\displaystyle \int_{0}^{4}{(x+e^{2x})dx}$$ 


SOLUTION
$$\begin{array}{l} \int _{ 0 }^{ 4 }{ \left( { x+{ e^{ 2x } } } \right) dx }  \\ \Rightarrow \int _{ 0 }^{ 4 }{ xdx } +\int _{ 0 }^{ 4 }{ { e^{ 2x } }dx }  \\ \Rightarrow \dfrac { 1 }{ 2 } \left[ { { x^{ 2 } } } \right] _{ 0 }^{ 4 }+\dfrac { 1 }{ 2 } \left[ { { e^{ 2x } } } \right] _{ 0 }^{ 4 }+c \\ \Rightarrow \dfrac { 1 }{ 2 } \times 16=\dfrac { 1 }{ 2 } \left[ { { e^{ 8 } }-1 } \right] +c \\ 8+\dfrac { 1 }{ 2 } \left[ { { e^{ 8 } }-1 } \right] +c. \end{array}$$
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Subjective Medium Published on 17th 09, 2020
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