<s>[INST] Finding the area between a line and a curve 
The two equations are $x+1$ and $4x-x^2-1$.
The answer is $\frac{1}{6}$, but I've done it 4 different times and gotten -$\frac{15}{2}$ each time.
My working:


*

*$x+1$ = $4x-x^2-1$

*$x^2-3x+2 = 0$

*$(x-1)(x-2)$ means $x=1$ or $x=2$

*$\int_1^2$ $3x-x^2$

*$[\frac{3x^2}{2}-\frac{x^3}{3}]_1^2$

*$\frac{3(1)^2}{2}-\frac{(1)^3}{3}$ = $\frac{3}{2}-\frac{1}{3}$

*$\frac{3(2)^2}{2}-\frac{(2)^3}{3}$ = $\frac{12}{2}-\frac{8}{3}$

*($\frac{3}{2}-\frac{1}{3}$)-($\frac{12}{2}-\frac{8}{3}$) = $-\frac{9}{2}-\frac{9}{3}$

*-$\frac{15}{2}$

 [/INST] Step 5. onwards is not ok
$$\int_a^bf(x)  dx = F(b)-F(a) $$
$$\int_a^bf(x)  dx = F(b-a) $$
The first result is correct but second one is in general wrong.
EDIT1:
The constant term in $ (x^2-3 x +2) $ is missing for integration. Factorization is not beneficial,
</s>