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

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

1 Answer

7 votes

Answer:

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

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