Question Statement

Find: ${\int }_1^5[3f(x)-2g(x)]dx$ if ${\int }_1^{5}f(x)dx=4$ and ${\int }_1^{5}g(x)dx=5$

---

Background and Explanation

This problem applies the linearity property of definite integrals, which allows constants to be factored out and integrals to be distributed over addition and subtraction. This property is essential for breaking down complex integrals into simpler components.

---

Solution

To evaluate this integral, we apply the linearity properties of definite integrals. Specifically, we use two key rules:

1. Constant Multiple Rule: ${\int }_a^{b}c\cdot f(x)dx=c{\int }_a^{b}f(x)dx$
2. Difference Rule: ${\int }_a^b[f(x)-g(x)]dx={\int }_a^{b}f(x)dx-{\int }_a^{b}g(x)dx$

Applying these properties to separate the given integral into individual components:

$\begin{array}{rl}{\int }_1^5[3f(x)-2g(x)]dx& =3{\int }_1^{5}f(x)dx-2{\int }_1^{5}g(x)dx\\ & =3(4)-2(5)\\ & =12-10\\ & =2\end{array}$

Thus, the value of the definite integral is $\boxed{2}$.

---

Key Formulas or Methods Used

• Linearity of Integration: ${\int }_a^b[c_{1}f(x)+c_{2}g(x)]dx=c_1{\int }_a^{b}f(x)dx+c_2{\int }_a^{b}g(x)dx$
• Constant Multiple Rule: ${\int }_a^{b}k\cdot f(x)dx=k{\int }_a^{b}f(x)dx$ (where $k$ is a constant)
• Integral of a Difference: ${\int }_a^b[f(x)-g(x)]dx={\int }_a^{b}f(x)dx-{\int }_a^{b}g(x)dx$

---

Summary of Steps

1. Apply linearity property: Factor out the constants 3 and 2, splitting the single integral into a difference of two integrals: ${\int }_1^5[3f(x)-2g(x)]dx=3{\int }_1^{5}f(x)dx-2{\int }_1^{5}g(x)dx$

2. Substitute given values: Replace ${\int }_1^{5}f(x)dx$ with 4 and ${\int }_1^{5}g(x)dx$ with 5: $3(4)-2(5)$

3. Evaluate the expression: Calculate $12-10=2$