given a function f(x) , find a g(x) such that g'(x)=f(x)
(1) 1/(4+x^2)
(2) cos x * sin x
great appreciate for any help
They do anti-derivatives in Calculus 1, in universities. They usually don't learn about Riemann sums and such until the next course in Calculus. These question were probably assigned or done in lecture and he doesn't understand them.
If your given a function $\displaystyle f(x) $ we're basically trying to find a function $\displaystyle F(x) $ such that $\displaystyle F'(x) = f(x) $
Knowing this we can reverse some of the easier differential rules.
Like the power rule. if $\displaystyle f(x) = x^n$ then $\displaystyle F(x) = \frac{1}{n+1}x^{n+1} + C$
Since the derivative of a constant is 0.
Or with cos and sin. if $\displaystyle f(x) = \cos{x}$ then $\displaystyle F(x) = \sin{x} + C$
if $\displaystyle f(x) = \sin{x}$ then $\displaystyle F(x) = -\cos{x} + C$
For the first question note that: $\displaystyle \frac{d}{dx}(\arctan{\frac{x}{a}}) = \frac{1}{1 + \frac{x^2}{a^2}}$
For the second question, a helpful identity is that $\displaystyle \sin{2x} = 2\sin{x}\cos{x}$