f is a continuous function whose domain is all real numbers. a,b,c,d,m, and n are constants. ∫[b a]f(x)dx=8, ∫[c b]f(x)dx=-11, ∫[d c]f(x)dx=9. Evaluate the following definite integrals.

Question

∫[d a]f(x)dx=?

3∫[b a]f(x)dx-5 ∫[c b]f(x)dx=?

∫[d b]f(x)dx-∫[c d]f(x)dx=?

∫[a b]f(x)dx=?

∫[m b]f(x)dx+∫[n m]f(x)dx+∫[c n]f(x)dx=?

∫[b a]f(x)dx+∫[b c]f(x)dx=?

∫[d b]f(x)dx=?

∫[b b]f(x)dx=?

Answer 1

∫[a,b]f(x)dx = -8 , ∫[b,c]f(x)dx = 11, ∫[c,d]f(x)dx = -9. So ... ∫[a,d]f(x)dx = -6.

3∫[b a]f(x)dx-5 ∫[c b]f(x)dx= 3⋅8 - 5⋅(-11) = 79

∫[d b]f(x)dx-∫[c d]f(x)dx= 9 - 11 - (-9) = 7

∫[a b]f(x)dx= -8 (see above)

∫[m b]f(x)dx+∫[n m]f(x)dx+∫[c n]f(x)dx = ∫[c b]f(x)dx = -11

∫[b a]f(x)dx+∫[b c]f(x)dx = 8 + 11 = 19

∫[d b]f(x)dx = 9 - 11 = -2

∫[b b]f(x)dx = 0