Question
If \( \int_{0}^{7} f(x) d x=31 \) and \( \int_{0}^{7} g(x) d x=16 \), find \( \int_{0}^{7}[2 f(x)+3 g(x)] d x \) \( x \)
Ask by Bowers Sherman. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The integral \( \int_{0}^{7} [2 f(x) + 3 g(x)] \, dx \) equals 110.
Solution
To find the value of the integral \( \int_{0}^{7} [2 f(x) + 3 g(x)] \, dx \), we can use the linearity property of definite integrals. The linearity property allows us to separate and scale integrals as follows:
\[
\int_{a}^{b} [c_1 f(x) + c_2 g(x)] \, dx = c_1 \int_{a}^{b} f(x) \, dx + c_2 \int_{a}^{b} g(x) \, dx
\]
Given:
\[
\int_{0}^{7} f(x) \, dx = 31
\]
\[
\int_{0}^{7} g(x) \, dx = 16
\]
Applying the linearity property:
\[
\int_{0}^{7} [2 f(x) + 3 g(x)] \, dx = 2 \int_{0}^{7} f(x) \, dx + 3 \int_{0}^{7} g(x) \, dx
\]
Substitute the given values:
\[
= 2 \times 31 + 3 \times 16
\]
\[
= 62 + 48
\]
\[
= 110
\]
**Answer:** 110
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Simplify this solution Beyond the Answer
To find \( \int_{0}^{7}[2 f(x)+3 g(x)] d x \), we can use the linearity of the integral. This means we can break the integral into parts: \[ \int_{0}^{7}[2 f(x)+3 g(x)] d x = \int_{0}^{7} 2 f(x) \, d x + \int_{0}^{7} 3 g(x) \, d x \] Now, we factor out the constants: \[ = 2 \int_{0}^{7} f(x) \, d x + 3 \int_{0}^{7} g(x) \, d x \] Substituting the given values: \[ = 2(31) + 3(16) \] \[ = 62 + 48 = 110 \] Thus, \[ \int_{0}^{7}[2 f(x)+3 g(x)] d x = 110 \]
