`int_0^ln2 2e^(-x)coshx dx` Evaluate the integral

Friday, March 6, 2015

$\displaystyle{\int_{{0}}^{{\ln{{2}}}}}{2}{{e}}^{{-{x}}}{\cosh{{x}}}{\left.{d}{x}\right.}$ Evaluate the integral

To compute this integral, recall the definition of hyperbolic cosine function: $\displaystyle{\cosh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}.$

Therefore the function under integral is equal to $\displaystyle{1}+{{e}}^{{-{2}{x}}},$ and its indefinite integral is $\displaystyle{x}-\frac{{1}}{{2}}{{e}}^{{-{2}{x}}}+{C}.$

Now we can substitute the given limits of integration to the antiderivative and obtain

$\displaystyle{\int_{{0}}^{{{\ln{{2}}}}}}{2}{{e}}^{{-{x}}}{\cosh{{\left({x}\right)}}}{\left.{d}{x}\right.}={\ln{{2}}}-\frac{{1}}{{2}}{\left({{e}}^{{-{2}{\ln{{2}}}}}\right)}-{\left({0}-\frac{{1}}{{2}}\right)}=$

$\displaystyle={\ln{{2}}}-\frac{{1}}{{8}}+\frac{{1}}{{2}}=$ ln2 + 3/8

(we used the identity $\displaystyle{{e}}^{{{\ln{{y}}}}}={y}$ and the property of exponent $\displaystyle{{\left({{a}}^{{b}}\right)}}^{{c}}={{a}}^{{{b}{c}}}$ ).

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Friday, March 6, 2015

$\displaystyle{\int_{{0}}^{{\ln{{2}}}}}{2}{{e}}^{{-{x}}}{\cosh{{x}}}{\left.{d}{x}\right.}$ Evaluate the integral

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, March 6, 2015

Evaluate the integral

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Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{\int_{{0}}^{{\ln{{2}}}}}{2}{{e}}^{{-{x}}}{\cosh{{x}}}{\left.{d}{x}\right.}$ Evaluate the integral

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?