Rewrite each of the following integral expressions as single integrals.

1. $\int _ { - 3 } ^ { - 5 } f ( x ) d x + \int _ { - 5 } ^ { - 3 } g ( x ) d x$

   Hint (a):

   $\int_{a}^{b}f(x)dx=\int_{b}^{a}-f(x)dx$

   Answer (a):

   $\int_{-5}^{-3}\left (g(x)-f(x) \right )dx$

2. $3 \int _ { 1 } ^ { 6 } f ( x ) d x + 5 \int _ { 1 } ^ { 6 } g ( x ) d x$

   Hint (b):

   Translation: three times the area under $f\left(x\right)$ between $x = 1$ and $x = 6 +\text{ five times the same area}$ . Write this as a single integral.

3. $\int _ { 6 } ^ { 11 } f ( x ) d x + \int _ { 11 } ^ { 6 } f ( x ) d x$

   Hint (c):

   See hint (a) and then see hint (c).

4. $\int _ { 7 } ^ { 10 } f ( t ) d t - \int _ { 7 } ^ { 9 } f ( t ) d t$

   Hint (d):

   Since an integral is the area underneath or above a curve, if you take a smaller portion of that area and subtract it from a larger portion, both starting at the same point, what are you left with?