ตอนที่2
9)$f(x+y)=f(x)+f(y)+xy-50,f(1)=5\rightarrow f(101)=?$
$f(n)=f(n-1)+f(1)+(1)(n-1)-50=f(n-2)+2f(1)+(1)((n-2)+(n-1))-2(50)=...=5n+\dfrac{(n-100)(n-1)}{2}$
$\therefore f(101)=5(101)+\dfrac{(1)(100)}{2}=555$ ข้อนี้ถึงคำตอบจะขำแต่คนที่คิดไม่ออกอาจจะไม่ขำนะครับ
10)$\displaystyle{\left(a-\frac{1}{c}\right)\left(b-\frac{1}{a}\right)\left(c-\frac{1}{b}\right)=\frac{(abc)^2-abc(a+b+c)+(ab+bc+ca)-1}{abc}=n}$
$\displaystyle{\therefore\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{abc}=1}$ at this point let consider $\displaystyle{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1}$ where $\displaystyle{\frac{1}{6}=\frac{1}{5}-\frac{1}{(2)(3)(5)}}$
$\therefore a^2+b^2+c^2=4+9+25=38$
__________________
$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$
BUT
$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$