Question

Given \int_{2}^{5}f(x)dx=17 and \int_{2}^{5}g(x)dx=-2, evaluate the following. (a)\int_{2}^{5}[f(x)+g(x)]dx (b)\int_{2}^{5}[g(x)-f(x)]dx (c)\int_{2}^{5}2g(x)dx (d)\int_{2}^{5}3f(x)dx

Applications of integrals
ANSWERED
asked 2021-06-05
Given \(\int_{2}^{5}f(x)dx=17\) and \(\int_{2}^{5}g(x)dx=-2\), evaluate the following.
(a)\(\int_{2}^{5}[f(x)+g(x)]dx\)
(b)\(\int_{2}^{5}[g(x)-f(x)]dx\)
(c)\(\int_{2}^{5}2g(x)dx\)
(d)\(\int_{2}^{5}3f(x)dx\)

Answers (1)

2021-06-06
Step 1
According to the standard rules of definite integrals, we have:
1.\(\int_{a}^{b}[f(x)\pm g(x)]dx=\int_{a}^{b}f(x)dx\pm \int_{a}^{b}g(x)dx\)
2.\(\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx\)
Step 2
(a)
\(\int_{2}^{5}[f(x)+g(x)]dx=\int_{2}^{5}f(x)dx+\int_{2}^{5}g(x)dx=(17)+(-2)=15\)
(b)
\(\int_{2}^{5}[g(x)-f(x)]dx=\int_{2}^{5}g(x)dx-\int_{2}^{5}f(x)dx=(-2)-(17)=-19\)
(c)
\(\int_{2}^{5}2g(x)dx=2\int_{2}^{5}g(x)dx=2(-2)=-4\)
(d)
\(\int_{2}^{5}3f(x)dx=3\int_{2}^{5}f(x)dx=3(17)=51\)
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