New Notes Blog: `int_0^pi (5e^x + 3sin(x))dx` Evaluate the integral

Saturday, 29 August 2015

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$ Evaluate the integral

You need to evaluate the definite integral using the fundamental theorem of calculus, such that: $\displaystyle{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({b}\right)}-{F}{\left({a}\right)}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={\int_{{0}}^{\pi}}{5}{{e}}^{{x}}{\left.{d}{x}\right.}+{\int_{{0}}^{\pi}}{3}{\sin{{x}}}{\left.{d}{x}\right.}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={\left({5}{{e}}^{{x}}-{3}{\cos{{x}}}\right)}{{\mid}_{{0}}^{\pi}}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}-{3}{\cos{\pi}}-{5}{{e}}^{{0}}+{3}{\cos{{0}}}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}-{3}\cdot{\left(-{1}\right)}-{5}+{3}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}+{1}$

Hence,...

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={\left({5}{{e}}^{{x}}-{3}{\cos{{x}}}\right)}{{\mid}_{{0}}^{\pi}}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}-{3}{\cos{\pi}}-{5}{{e}}^{{0}}+{3}{\cos{{0}}}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}-{3}\cdot{\left(-{1}\right)}-{5}+{3}$

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}+{1}$

Hence, evaluating the definite integral yields

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{x}}}\right)}{\left.{d}{x}\right.}={5}{{e}}^{\pi}+{1}$

New Notes Blog

Saturday, 29 August 2015

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$ Evaluate the integral

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Saturday, 29 August 2015

Evaluate the integral

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In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

$\displaystyle{\int_{{0}}^{\pi}}{\left({5}{{e}}^{{x}}+{3}{\sin{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$ Evaluate the integral

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?