Question

Continuous functions f, g are known to have the properties z 6 2 f(x) dx = 12, z 6 2 g(x) dx = 24 respectively. Use these to find the value of the integral i = z 6 2 (6f(x) − 2g(x) + 5) dx.

Answer 1

we  are given

$\int _2^6f\left(x\right)dx=12\:$

$\int _2^6g\left(x\right)dx=24\:$

we can find

$\int _2^6(6f(x)-2g(x) +5)dx\:$

we can distribute it

$=6\int _2^6 f(x)dx-2\int _2^6 g(x)dx +\int _2^6 5dx$

we can plug values

and we get

$=6*12-2*24 +\int _2^6 5dx$

$\int _2^6\:5dx$

$\int \:5dx$

$=5x$

now, we can plug bounds

and we get

$=30-10$

$=20$

$=6*12-2*24 +20$

$=44$................Answer