43. let ( f ) be a function such that ( int_{6}^{12} f(2x)dx = 10 ). which of the following must be true? (…

43. let ( f ) be a function such that ( int_{6}^{12} f(2x)dx = 10 ). which of the following must be true? ( \bigcirc int_{12}^{24} f(t)dt = 5 ) ( \bigcirc int_{12}^{24} f(t)dt = 20 ) ( \bigcirc int_{6}^{12} f(t)dt = 5 ) ( \bigcirc int_{6}^{12} f(t)dt = 20 ) ( \bigcirc int_{3}^{6} f(t)dt = 5 )

43. let ( f ) be a function such that ( int_{6}^{12} f(2x)dx = 10 ). which of the following must be true? ( \bigcirc int_{12}^{24} f(t)dt = 5 ) ( \bigcirc int_{12}^{24} f(t)dt = 20 ) ( \bigcirc int_{6}^{12} f(t)dt = 5 ) ( \bigcirc int_{6}^{12} f(t)dt = 20 ) ( \bigcirc int_{3}^{6} f(t)dt = 5 )

Answer

Explanation:

Step1: Use substitution method

Let (t = 2x), then (dt=2dx), or (dx=\frac{1}{2}dt). When (x = 6), (t=2\times6 = 12); when (x = 12), (t=2\times12=24). So (\int_{6}^{12}f(2x)dx=\int_{12}^{24}f(t)\times\frac{1}{2}dt).

Step2: Solve for (\int_{12}^{24}f(t)dt)

Since (\int_{6}^{12}f(2x)dx = 10), and (\int_{6}^{12}f(2x)dx=\frac{1}{2}\int_{12}^{24}f(t)dt). We have (10=\frac{1}{2}\int_{12}^{24}f(t)dt). Multiply both sides by (2): (\int_{12}^{24}f(t)dt = 2\times10=20).

Answer:

(\int_{12}^{24}f(t)dt = 20) (the second option)