Question

If \(\int_0^9 f (x)dx = 37\) and \(\int_0^9 g (x)dx = 16\), find\(\int_0^9 {(2f(} x) + 3g(x))dx\)

Short answer

The value of the integral \(\int_0^9 {(2f(} x) + 3g(x))dx\) is \(122\).

Step-by-step solution

Given information

The integral function\(\int_0^9 {(2f(} x) + 3g(x))dx\).

The value of the integrals are\(\int_0^9 f (x)dx = 37\)and\(\int_0^9 g (x)dx = 16\).

Calculation of the integral

Consider the integral function\(\int_0^9 {(2f(} x) + 3g(x))dx{\kern 1pt} {\kern 1pt} \,{\kern 1pt} {\kern 1pt} .......(1)\)

Apply property of integral

\(\int_a^b c f(x)dx = c\int_a^b f (x)dx{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} .........(2)\)

Here,\(c\)is a constant.

Modify the Equation (1) using Equation (2).

\(\int_0^9 {(2f(} x) + 3g(x))dx = 2\int_0^9 f (x)dx + 3\int_0^9 g (x)dx{\kern 1pt} {\kern 1pt} {\kern 1pt} \,{\kern 1pt} ..........(3)\)

Substitute the value in the integrals

Substitute 37 for\(\int_0^9 f (x)dx\)and 16 for\(\int_0^9 f (x)dx\)in Equation (2).

\(\begin{aligned}{l}\int_0^9 {(2f(} x) + 3g(x))dx &= 2\int_0^9 f (x)dx + 3\int_0^9 g (x)dx\\ &= (2 \times 37) + (3 \times 16)\\ &= 74 + 48\\ &= 122\end{aligned}\)

Hence, the value of the integral function is \(122 .\)