Question

Suppose integral subscript 3 superscript 5 f (x )space d x equals 5 integral subscript 3 superscript 9 g (x )space d x equals 3 space integral subscript 5 superscript 9 g (x )space d x space equals 4 Find the value of integral subscript 3 superscript 5 open parentheses f (x )plus 2 g (x )close parentheses space d x

Answer 1

Answer:

  3

Step-by-step explanation:

First we need to find the integral from 3 to 5 of g(x).

  $\int\limits^9_3 {g(x)} \, dx =\int\limits^5_3 {g(x)} \, dx +\int\limits^9_5 {g(x)} \, dx \\\\\int\limits^5_3 {g(x)} \, dx =\int\limits^9_3 {g(x)} \, dx -\int\limits^9_5 {g(x)} \, dx\\\\\int\limits^5_3 {g(x)} \, dx =3-4=-1$

Then the integral of interest is ...

  $\int\limits^5_3 {(f(x)+2g(x))} \, dx =\int\limits^5_3 {f(x)} \, dx +2\int\limits^5_3 {g(x)} \, dx =5+2(-1)\\\\\boxed{\int^5_3 {(f(x)+2g(x))} \, dx =3}$