Solving Integral Issues: "If b∫f(x)dx = a + 2b, then ∫ (f(x) + 5)dx?

[kenny87]

Here's the question:

If:

b b
∫ f(x)dx = a + 2b, then ∫ (f(x) + 5)dx = ?
a a

I'm thinking myself into circles... I want to say I need to take the derivative of a+2b to then find out what equals f(x) and then just take the integral of that +5... but its just not working out.

 

[Tide]

If I am reading correctly what you wrote then you are merely adding 5(b-a) to the original integral.

 

[HallsofIvy]

$$\int_a^b (f(x)+ 5)dx= \int_a^b f(x)dx+ 5\int_a^b dx$$

 

[kenny87]

So would it be just a+2b+5?

 

[Physics Monkey]

Nope, what does $$\int^b_a dx$$ equal?

 

[kenny87]

Ok, so then I just do 5(b-a)?

 

[TD]

You have to add that, yes

 

[kenny87]

Yeah, that's what I meant to say.

So in this problem:

If f(x)=g(x)+7 from 3 to 5, then the integral from 3 to 5 of [f(x)+g(x)]dx is?

Can I just use the same method and get

5
2 ∫ g(x)dx+7
3

 

[TD]

Almost, don't forget that the 7 was in the integrand!

$$\int\limits_3^5 {f\left( x \right) + g\left( x \right)dx} = \int\limits_3^5 {g\left( x \right) + 7 + g\left( x \right)dx} = 2\int\limits_3^5 {g\left( x \right)dx} + 7\int\limits_3^5 {dx}$$

 

[kenny87]

how do i figure dx in this case? do i use g(x) or f(x)?

 

[TD]

You either use f(x) and substitute g(x) by f(x)-7 or you use g(x), and substitute f(x) by g(x)+7.