# Equating two integrals with a constant involved

1. Dec 6, 2016

### Arnoldjavs3

1. The problem statement, all variables and given/known data
$\int_{-1}^{3} f(x) dx = -4 = \int_{-1}^{3} 2g(x)dx$

Now find a value(constant a) that makes the following true:

$\int_{-1}^{3} [3f(x) - ag(x) +a] dx = \int_{-1}^{3}(1-ax)dx$

2. Relevant equations

3. The attempt at a solution
I'm unsure if my approach here is correct but I think that I need to utilize the fact that $\int_{-1}^{3} f(x) dx$ = -4 and substitute this in to the left side of equation? So I would get something like this after evaluating:

$\int_{-1}^{3} [-12+2a+a] dx$ and then once i've done this, i can solve the integral. I'll have to solve the integral on the right side as well and once i've finished that, I find a value for a? I got 52/16 but I dont have an answer to check to.

2. Dec 6, 2016

### Staff: Mentor

You can also use the fact that $\int_{-1}^{3} 2g(x)dx = -4$ to get an expression for $\int_{-1}^{3} g(x)dx$.
The two properties of definite integrals that come into play here are:
$\int_a^b k \cdot f(x) dx = k\int_a^b f(x) dx$ (k a constant)
and $\int_a^b f(x) + g(x) dx = \int_a^b f(x) dx + \int_a^b g(x) dx$

3. Dec 6, 2016

### Arnoldjavs3

using these properties would I get something like this?(after splitting the left side):
$3\int_{-1}^{3} f(x)dx - a\int_{-1}^{3} g(x)dx + \int_{-1}^{3} adx$ ?

4. Dec 6, 2016

### Staff: Mentor

Yes, and you should be able to evaluate all three of these integrals, based on the given information.