Question

To find ∫23[6f(x)−g(x)]dx given ∫23=2∫23f(x)dx=2 and ∫23=8∫23g(x)dx=8, you can apply linearity of integrals to find ∫236f(x)dx and ∫23g(x)dx separately, and then subtract.

Answer 1

To find ∫23[6f(x)−g(x)]dx, you can use the linearity property of integrals. Given the equations ∫23 = 2∫23f(x)dx = 2 and ∫23 = 8∫23g(x)dx = 8, you can find the values of ∫236f(x)dx and ∫23g(x)dx separately, and then subtract them to get the final result.

Step-by-step explanation:

To find ∫23[6f(x)−g(x)]dx, we can use the linearity property of integrals. We are given that ∫23 = 2∫23f(x)dx = 2 and ∫23 = 8∫23g(x)dx = 8. Using these equations, we can find the values of ∫236f(x)dx and ∫23g(x)dx separately, and then subtract them.

Step 1: Substitute the given equations into the first equation of ∫23[6f(x)−g(x)]dx.
∫23[6f(x)−g(x)]dx = 2[6f(x)−g(x)]dx = 12f(x)dx − 2g(x)dx

Step 2: Use the given equations, ∫23 = 2 and ∫23 = 8, to calculate the values of ∫236f(x)dx and ∫23g(x)dx.

∫236f(x)dx = 2 * 6f(x)dx = 12f(x)dx

∫23g(x)dx = 8 * g(x)dx

Step 3: Substitute the calculated values into the equation from Step 1 and simplify.
∫23[6f(x)−g(x)]dx = 12f(x)dx − 2g(x)dx = ∫236f(x)dx − ∫23g(x)dx = 12f(x)dx − 8g(x)dx = 4f(x)dx − 8g(x)dx

So, ∫23[6f(x)−g(x)]dx = 4f(x)dx − 8g(x)dx.