Question

the values of several definite integrals are given as follows:
\\(\int_{0}^{2} f(x) dx = 5\\) \\(\int_{0}^{3} f(x) dx = 7\\) \\(\int_{0}^{2} g(x) dx = -3\\) \\(\int_{2}^{3} g(x) dx = 5\\)
use these values and properties of definite integrals to evaluate the indicated definite integral.
find a formula for \\(a\\) in terms of \\(b\\) such that \\(\int_{0}^{3} (a f(x) + b g(x)) dx = 0\\).

Explanation:

Step1: Use linearity property of definite integrals

$$\int_{0}^{3}(af(x)+bg(x))dx=a\int_{0}^{3}f(x)dx + b\int_{0}^{3}g(x)dx$$
We know that \(\int_{0}^{3}g(x)dx=\int_{0}^{2}g(x)dx+\int_{2}^{3}g(x)dx\). Substituting the given values \(\int_{0}^{2}g(x)dx=- 3\) and \(\int_{2}^{3}g(x)dx = 5\), we get \(\int_{0}^{3}g(x)dx=-3 + 5=2\)

Step2: Substitute the values of \(\int_{0}^{3}f(x)dx\) and \(\int_{0}^{3}g(x)dx\)

Since \(\int_{0}^{3}(af(x)+bg(x))dx = 0\), and \(\int_{0}^{3}f(x)dx = 7\), \(\int_{0}^{3}g(x)dx=2\)
We have \(a\times7+b\times2 = 0\)

Step3: Solve for \(a\) in terms of \(b\)

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Answer:

\(a =-\frac{2}{7}b\)