Question

if $int_{-1}^{1}f(x)dx = 7$ and $int_{-1}^{1}g(x)dx=-9$, then $int_{-1}^{1}(2f(x)-g(x))dx=$
(a) 5
(b) 16
(c) 23
(d) 32

Explanation:

Step1: Apply integral - linearity rule

By the linearity of definite integrals, $\int_{a}^{b}(cf(x)\pm dg(x))dx = c\int_{a}^{b}f(x)dx\pm d\int_{a}^{b}g(x)dx$, where $c$ and $d$ are constants. Here, $a=-1$, $b = 1$, $c = 2$ and $d=-1$. So, $\int_{-1}^{1}(2f(x)-g(x))dx=2\int_{-1}^{1}f(x)dx-\int_{-1}^{1}g(x)dx$.

Step2: Substitute given values

We know that $\int_{-1}^{1}f(x)dx = 7$ and $\int_{-1}^{1}g(x)dx=-9$. Substitute these values into the above - derived formula: $2\times7-(-9)$.

Step3: Calculate the result

First, $2\times7 = 14$, and $-(-9)=9$. Then $14 + 9=23$.

Answer:

C. 23