Blog: If the integral from 1 to 4 of f(x)=7, then what does the integral from 1 to 4 of 2f(x)+5=?

Monday, October 10, 2016

If the integral from 1 to 4 of f(x)=7, then what does the integral from 1 to 4 of 2f(x)+5=?

$\displaystyle{\int_{{1}}^{{4}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={7}$

Find $\displaystyle{\int_{{1}}^{{4}}}{\left({2}{f{{\left({x}\right)}}}+{5}\right)}{\left.{d}{x}\right.}$ .

To solve, express this as sum of two integrals.

$\displaystyle{\int_{{1}}^{{4}}}{\left({2}{f{{\left({x}\right)}}}+{5}\right)}{\left.{d}{x}\right.}$

$\displaystyle={\int_{{1}}^{{4}}}{2}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}+{\int_{{1}}^{{4}}}{5}{\left.{d}{x}\right.}$

Then, factor out the 2 in the first integral.

$\displaystyle={2}{\int_{{1}}^{{4}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}+{\int_{{1}}^{{4}}}{5}{\left.{d}{x}\right.}$

Plug-in the given $\displaystyle{\int_{{1}}^{{4}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={7}$ .

$\displaystyle={2}{\left({7}\right)}+{\int_{{1}}^{{4}}}{5}{\left.{d}{x}\right.}$

$\displaystyle={14}+{\int_{{1}}^{{4}}}{5}{\left.{d}{x}\right.}$

For the second integral, apply the formula int c dx = cx.

$\displaystyle={14}+{5}{x}{{\mid}_{{1}}^{{4}}}$

To evaluate the definite integral, apply the formula $\displaystyle{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={F}{\left({b}\right)}-{F}{\left({a}\right)}$ .

$\displaystyle={14}+{\left({5}\cdot{4}-{5}\cdot{1}\right)}$

$\displaystyle={14}+{15}$

$\displaystyle={29}$

Therefore, $\displaystyle{\int_{{1}}^{{4}}}{\left({2}{f{{\left({x}\right)}}}+{5}\right)}{\left.{d}{x}\right.}={29}$ .

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Monday, October 10, 2016

If the integral from 1 to 4 of f(x)=7, then what does the integral from 1 to 4 of 2f(x)+5=?

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