Evaluate Definite Integral Using Integral Properties: If \int_{1}^{7}f(x)dx=2.5

nitraiddQ 2021-08-20 Answered
Evaluate Definite Integral Using Integral Properties:
If \(\displaystyle{\int_{{{1}}}^{{{7}}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={2.5}\) and \(\displaystyle{\int_{{{1}}}^{{{7}}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}={4}\), find \(\displaystyle{\int_{{{1}}}^{{{7}}}}{\left[{4}{f{{\left({x}\right)}}}-{2}{g{{\left({x}\right)}}}\right]}{\left.{d}{x}\right.}\).

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Expert Answer

hosentak
Answered 2021-08-21 Author has 18699 answers
Step 1
\(\displaystyle{\int_{{{1}}}^{{{7}}}}{\left[{4}{f{{\left({x}\right)}}}-{2}{g{{\left({x}\right)}}}\right]}{\left.{d}{x}\right.}={\int_{{{1}}}^{{{7}}}}{4}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}-{\int_{{{1}}}^{{{7}}}}{2}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\)
\(\displaystyle={4}{\int_{{{1}}}^{{{7}}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}-{2}{\int_{{{1}}}^{{{7}}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\)
\(\displaystyle={4}{\left({2.5}\right)}-{2}{\left({4}\right)}\)
\(\displaystyle={10}-{8}\)
\(\displaystyle={2}\)
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