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