If the integral from 1 to 4 of f(x)=7, then what does the integral from 1 to 4 of 2f(x)+5=?

`int_1^4 f(x) dx= 7`

Find `int_1^4 (2f(x) + 5)dx` .

To solve, express this as sum of two integrals.

`int_1^4 (2f(x) + 5)dx`

`=int_1^4 2f(x) dx + int_1^4 5dx`

Then, factor out the 2 in the first integral.

`= 2int_1^4 f(x) dx + int_1^4 5dx`

Plug-in the given `int_1^4 f(x) dx= 7` .

`=2(7) + int_1^4 5 dx`

`= 14 + int_1^4 5 dx`

For the second integral, apply the formula int c dx = cx.

`= 14 + 5x|_1^4`

To evaluate the definite integral, apply the formula `int_a^b f(x) dx = F(b) - F(a)` .

`= 14 + (5*4 - 5*1)`

`=14 + 15`

`= 29`

Therefore,  `int_1^4 (2f(x) + 5)dx = 29` .