Question

Find the value of integral(4 to 10) 2f(x)-3dx

Answer 1

We know that

$\displaystyle \int_1^4 f(x) \, dx = 8$

$\displaystyle \int_1^{10} f(x) \, dx = 17$

By linearity of the definite integral,

$\displaystyle \int_4^{10} 2f(x) - 3 \, dx = 2 \int_4^{10} f(x) \, dx - 3 \int_4^{10} dx$

Presumably, you're aware that

$\displaystyle \int_a^b dx = b - a$

for any two numbers a and b, so the last integral is simply 6.

From the known integrals, we also know

$\displaystyle \int_1^{10} f(x) \, dx = \int_1^4 f(x) \, dx + \int_4^{10} f(x) \, dx$

$\displaystyle 17 = 8 + \int_4^{10} f(x) \, dx$

$\displaystyle \int_4^{10} f(x) \, dx = 9$

so

$\displaystyle \int_4^{10} 2f(x) - 3 \, dx = 2\times9- 3\times6 = \boxed{0}$