Area Between 2 Curves Integration Help

Question

If my two functions are $$f(x)=\frac{6x}{x^2+1}$$ and $$g(x)=\frac{3x}{5}$$, I solved for the intersection points, $x=3,0,-3$, and drew a rough sketch of the function. Would my next step to be to integrate the area from $-3$ to $0$, and then add that value to the value of when I integrate $0$ to $3$?

Answer 1

Yes, you get the right idea.

Also, take advantage of symmetry, the first piece of integral is equal to the second one.

Just evaluate

$$2\int_0^3 \frac{6x}{x^2+1}-\frac{3x}{5}\, dx$$

Answer 2

you can integrate $$2\int_{0}^3 \left(\frac{6x}{x^2+1}-\frac{3x}{5}\right)dx$$ the result should be $$-{\frac {27}{5}}+6\,\ln \left( 2 \right) +6\,\ln \left( 5 \right) $$