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### Section 5.6 : Definition of the Definite Integral

5. Determine the value of $$\displaystyle \int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}}$$ given that $$\displaystyle \int_{6}^{{11}}{{f\left( x \right)\,dx}} = - 7$$ and $$\displaystyle \int_{6}^{{11}}{{g\left( x \right)\,dx}} = 24$$.

Show Solution
There really isn’t much to this problem other than use the properties from the notes of this section until we get the given intervals at which point we use the given values.

\begin{align*}\int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}} & = \int_{6}^{{11}}{{6g\left( x \right)\,dx}} - \int_{6}^{{11}}{{10f\left( x \right)\,dx}} & \hspace{0.5in} & {\mbox{Property 4}}\\ & = 6\int_{6}^{{11}}{{g\left( x \right)\,dx}} - 10\int_{6}^{{11}}{{f\left( x \right)\,dx}} & \hspace{0.5in} & {\mbox{Property 3}}\\ & = 6\left( {24} \right) - 10\left( { - 7} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{214}} & & \end{align*}