Question

Let f be a function such that f(x)dx=5 and f(x)dx=1 What is the value of 3f(x)dx?

Answer 1

A) 12

Explanation:

Notice that
`\int\limits^5_1 {f(x)} \, dx=-\int\limits^1_5 {f(x)} \, dx` by the exchange-of-limits property of integrals, so
`-\int\limits^1_5 {f(x)} \, dx=-1`

We also know that
`\int\limits^1_(-1) {f(x)} \, dx+\int\limits^5_1 {f(x)} \, dx=\int\limits^5_(-1) {f(x)} \, dx` by the additivity property of integrals, which means that:

`\int\limits^1_(-1) {f(x)} \, dx+\int\limits^5_1 {f(x)} \, dx=\int\limits^5_(-1) {f(x)} \, dx\\\\\int\limits^1_(-1) {f(x)} \, dx-\int\limits^1_5 {f(x)} \, dx=\int\limits^5_(-1) {f(x)} \, dx\\\\5-1=\int\limits^5_(-1) {f(x)} \, dx\\\\4=\int\limits^5_(-1) {f(x)} \, dx`

Therefore,
`3\int\limits^5_(-1) {f(x)} \, dx=3(4)=12`