# Evaluating integrals Evaluate the following integral. \int_{0}^{\ln 2}\int_{e^{x}}^{2}dy dx

Evaluating integrals Evaluate the following integral.
${\int }_{0}^{\mathrm{ln}2}{\int }_{{e}^{x}}^{2}dydx$
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Step 1
Concept:
The calculus helps in understanding the changes between values that are related by a function. It is used in different areas such as physics, biology and sociology and so on. The calculus is the language of engineers, scientists and economics.
Step 2
Given:
${\int }_{0}^{\mathrm{ln}2}{\int }_{{e}^{x}}^{2}dydx$
the objective is to integrate the given integral.
Step 3
${\int }_{0}^{\mathrm{ln}2}{\int }_{{e}^{x}}^{2}dydx$
On integrating
${\int }_{{e}^{x}}^{2}1dt=2-{e}^{x}$
${\int }_{0}^{\mathrm{ln}2}{\int }_{{e}^{x}}^{2}1dydx={\int }_{0}^{\mathrm{ln}2}\left(2-{e}^{x}\right)dx$
${\int }_{0}^{\mathrm{ln}2}\left(2-{e}^{x}\right)dx=2\mathrm{ln}\left(2\right)-1$
${\int }_{0}^{\mathrm{ln}2}{\int }_{{e}^{x}}^{2}1dydx=2\mathrm{ln}\left(2\right)-1$