Question

Continuous functions f, g are known to have the properties z 4 1 f(x) dx = 19, z 4 1 g(x) dx = 12 respectively. use these to find the value of the integral i = z 4 1 (8f(x) − g(x) + 7) dx. 1. i = 157 2. i = 161 3. i = 159 4. i = 163 5. i = 155

Answer 1

I guess this reads

$\displaystyle\int_1^4f(x)\,\mathrm dx=19$

$\displaystyle\int_1^4g(x)\,\mathrm dx=12$

You want to compute

$I=\displaystyle\int_1^4(8f(x)-g(x)+7)\,\mathrm dx$

By linearity of the definite integral,

$I=\displaystyle8\int_1^4f(x)\,\mathrm dx-\int_1^4g(x)\,\mathrm dx+7\int\mathrm dx$

$I=8\cdot19-12+7(4-1)=\boxed{161}$

Answer 2

The value of the integral is 161.

We have

$\int\limits^4_1 {f(x)} \, dx =19\\\int\limits^4_1 {g(x)} \, dx =12$

Now, the integral

$\int\limits^4_1 {8f(x)-g(x)+7} \, dx \\=\int\limits^4_1 {8f(x)}dx-\int\limits^4_1g(x)+\int\limits^4_17} \, dx\\=8\int\limits^4_1 {f(x)}dx-\int\limits^4_1g(x)+7\int\limits^4_1} \, dx\\=8\times19-12+7(4-1)\\=8\times19-12+21\\=161$

The value of the integral $\int\limits^4_1 {8f(x)-g(x)+7} \, dx$ 161.

Therefore the correct option is 4.161

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