Additivity Integration Problem

[rrrright]

Homework Statement

View attachment latex-image-1.pdf equals...

Homework Equations

$$\int_a^{b} f(x)dx+\int_b^{c} f(x)dx=\int_a^{c} f(x)dx$$

$$\int_b^{a} f(x)dx=-\int_a^{b} f(x)dx$$

The Attempt at a Solution

View attachment latex-image-1.pdf

View attachment latex-image-2.pdf

$$\int_2^{-1} f(x)dx-\int_2^{5} f(x)dx$$

$$\int_2^{-1} f(x)dx+\int_5^{2} f(x)dx$$

$$\int_5^{-1} f(x)dx$$

The answer key says the answer is View attachment latex-image-3.pdf Where did i go wrong?

 

[Mark44]

Homework Statement

View attachment 30603 equals...

Homework Equations

$$\int_a^{b} f(x)dx+\int_b^{c} f(x)dx=\int_a^{c} f(x)dx$$

$$\int_b^{a} f(x)dx=-\int_a^{b} f(x)dx$$

The Attempt at a Solution

View attachment 30603

View attachment 30604

$$\int_2^{-1} f(x)dx-\int_2^{5} f(x)dx$$

$$\int_2^{-1} f(x)dx+\int_5^{2} f(x)dx$$

$$\int_5^{-1} f(x)dx$$

The answer key says the answer is View attachment 30605 Where did i go wrong?

I get
$$-\int_{-1}^{5} f(x)dx$$
which is the same as what you got. It looks to me like the answer key is missing the - sign out front.