Question

(06.04 MC)

If
`\int\limits^3_ {-2} \, [2f(x)+2]dx=18` and
`\int\limits^1_ {-2} \, f(x)dx =8`, then
`\int\limits^3_ {1} \, f(x)dx` is equal to which of the following?

4

0

−2

−4

Answer 1

`\huge\underline{\underline{\boxed{\mathbb {ANSWER:}}}}`

◉
`\large\bm{ -4}`

`\huge\underline{\underline{\boxed{\mathbb {SOLUTION:}}}}`

Before performing any calculation it's good to recall a few properties of integrals:

`\small\longrightarrow \sf{\int_(a)^b(nf(x) + m)dx = n \int^b _(a)f(x)dx + \int_(a)^bmdx}`

`\small\sf{\longrightarrow If \: a \angle c \angle b \Longrightarrow \int^(b) _a f(x)dx= \int^c _a f(x)dx+ \int^(b) _c f(x)dx }`

So we apply the first property in the first expression given by the question:

`\small \sf{\longrightarrow\int ^3_(-2) [2f(x) +2]dx= 2 \int ^3 _(-2) f(x) dx+ \int f^3 _(2) 2dx=18}`

And we solve the second integral:

`\small\sf{\longrightarrow2 \int ^3_(-2) f(x)dx + 2 \int ^3_(-2) f(x)dx = 2 \int ^3_(-2) f(x)dx + 2 \cdot(3 - ( - 2)) }`

`\small \sf{\longrightarrow 2 \int ^3_(-2) f(x)dx + 2 \int ^3_(-2) 2dx = 2 \int ^3_(-2) f(x)dx + 2 \cdot5 = 2 \int^3_(-2) f(x)dx10 = }`

Then we take the last equation and we subtract 10 from both sides:

`\sf{{\longrightarrow 2 \int ^3_(-2) f(x)dx} + 10 - 10 = 18 - 10}`

`\small \sf{\longrightarrow 2 \int ^3_(-2) f(x)dx = 8}`

And we divide both sides by 2:

`\small\longrightarrow \sf{\frac{2 { \int}^(3) _(2) }{2} = (8)/(2) }`

`\small \sf{\longrightarrow 2 \int ^3_(-2) f(x)dx=4}`

Then we apply the second property to this integral:

`\small \sf{\longrightarrow 2 \int ^3_(-2) f(x)dx + 2 \int ^3_(-2) f(x)dx + 2 \int ^3_(-2) f(x)dx = 4}`

Then we use the other equality in the question and we get:

`\small\sf{\longrightarrow 2 \int ^3_(-2) f(x)dx = 2 \int ^3_(-2) f(x)dx = 8 + 2 \int ^3_(-2) f(x)dx = 4}`

`\small\longrightarrow \sf{2 \int ^3_(-2) f(x)dx =4}`

We substract 8 from both sides:

`\small\longrightarrow \sf{2 \int ^3_(-2) f(x)dx -8=4}`

•
`\small\longrightarrow \sf{2 \int ^3_(-2) f(x)dx =-4}`