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

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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Step 1
$$\displaystyle{\int_{{{1}}}^{{{7}}}}{\left[{4}{f{{\left({x}\right)}}}-{2}{g{{\left({x}\right)}}}\right]}{\left.{d}{x}\right.}={\int_{{{1}}}^{{{7}}}}{4}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}-{\int_{{{1}}}^{{{7}}}}{2}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}$$
$$\displaystyle={4}{\int_{{{1}}}^{{{7}}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}-{2}{\int_{{{1}}}^{{{7}}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}$$
$$\displaystyle={4}{\left({2.5}\right)}-{2}{\left({4}\right)}$$
$$\displaystyle={10}-{8}$$
$$\displaystyle={2}$$