Calculus by Apostol Exercise 5.5 Problem 17

Question

There is a function $f$, defined and continuous for all real $x$, which satisfies the equation of the form $$\int_0^x f(t)\,dt = \int_x^1 t^2f(t)dt + \frac{x^{16}}{8} + \frac{x^{18}}{9} + c,$$

I tried to differentiate both sides giving me $f(t) = t^2 f(t) + 2x^{15} + 2x^{17}$, and $f(t) = (2x^{15}+2x^{17})/(1-x^2)$. The answer should be $2x^{15}$, I can't think of any way how to make it happen. Any help?

The derivative of $\int_{a}^{x}F(t)$ is $F(x)$. But then the derivative of $\int_{x}^{a}F(t)=-\int_{a}^{x}F(t)$ must be $-F(t)$. You didn't write that minus sign. — Alamos, Apr 21 '15 at 13:21

score 4 · Accepted Answer · answered Apr 21 '15 at 13:23

4

Note that

$$\frac{d}{dx}\int_{x}^{1}t^2f(t)dt=\color{red}{-}x^2f(x).$$ So, $$f(x)=-x^2f(x)+2x^{15}+2x^{17}\Rightarrow f(x)=\frac{2x^{15}(1+x^2)}{1+x^2}.$$

answered Apr 21 '15 at 13:23

mathlove

139,939

Oh, I see! Because the derivative of F(1) is zero so we are left with -x^2 f(x). Thanks! – shinobi20 Apr 21 '15 at 13:29
@shinobi20: Yes! You are welcome. – mathlove Apr 21 '15 at 13:29

score 2 · Answer 2 · answered Apr 21 '15 at 13:22

2

When you differentiate the equation: $$f(x)=\large{\color{#a00}{-}}x^2f(x)+2x^{15} + 2x^{17}$$ and you continue the calculation

answered Apr 21 '15 at 13:22

Elaqqad

13,725

How come f is a function of w on the left side and x on the right side, shouldn't it be f(x) = -x^2f(x) + 2x^15 +2x^17??? – shinobi20 Apr 21 '15 at 13:24
yes it's just atypo – Elaqqad Apr 21 '15 at 13:24

Calculus by Apostol Exercise 5.5 Problem 17

2 Answers2