If $\int_0^1f(x)\,dx=16$ and $\int_0^1g(x)\,dx=3.5$, then determine $\int_0^14\cdot f(x)-8\cdot g(x)\,dx$.

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Lesson Parent: 
Chapter 28 - Rules of Integrals
Problem: 

If $\int_0^1f(x)\,dx=16$ and $\int_0^1g(x)\,dx=3.5$, then determine $\int_0^14\cdot f(x)-8\cdot g(x)\,dx$.

Answer: 

$\int_0^14\cdot f(x)-8\cdot g(x)\,dx=36$

Solution: 

Use Integral Rule 4 which says
\[
\int_0^14\cdot f(x)-8\cdot g(x)\,dx=4\cdot \int_0^1f(x)\,dx-8\int_0^1g(x)\,dx.
\]
So this is nothing more than Algebra:
\begin{eqnarray}
\int_0^14\cdot f(x)-8\cdot g(x)\,dx&=&4\cdot \int_0^1f(x)\,dx-8\int_0^1g(x)\,dx\\
&=&4\cdot16-8\cdot 3.5\\
&=&64-28\\
&=&36
\end{eqnarray}