Hello to my lovely fellow math help forum posters,

I am in need of help with the following problems involving evaluating integrals. I took a picture of the problems that I need help with. It's 3 integral problems and 1 derivative problem involving max/min etc.

I would appreciate any help that can be given!!! I need help on the steps to this method.. step by step. Can someone write the steps out and take a picture of it and post it up because its easier than writing *int* instead of the integral sign? (Headbang)

Thanks (Nod)

Hint:

(1) For 1st integral: \(\displaystyle \int f'(x)f^2(x)\,dx=\frac{f^3(x)}{3}+C\,,\,\,f(x)\) any derivable function

(2) For 2nd integral: if \(\displaystyle \int f(x)\,dx=F(x)\,,\,\,then\,\,\,\int g'(x)f(g(x))\,dx=F(g(x))\) , with \(\displaystyle g(x)\) a derivable function ( note that here \(\displaystyle F(x)\) is the primitive function of \(\displaystyle f(x)\) )

(3) For 3rd integral: do integration by parts with \(\displaystyle u=x^2\Longrightarrow u'=2x\,,\,\,v'=e^{-x}\Longrightarrow v=-e^{-x}\)

(4) for 4th problem: a twice derivable function \(\displaystyle f(x)\) has:

(i) a maximum point at \(\displaystyle (x_0,f(x_0))\) if \(\displaystyle f'(x_0)=0\,,\,\,f''(x_0)<0\)

(ii) a minimum point at \(\displaystyle (x_0,f(x_0))\) if \(\displaystyle f'(x_0)=0\,,\,\,f''(x_0)>0\)

The function is increasing when \(\displaystyle f'(x)>0\) and decreasing when \(\displaystyle f'(x)<0\) , and concave upwards when \(\displaystyle f''(x)>0\) and concave downwards when \(\displaystyle f''(x)<0\) .

The function has an inflection point when the second derivative has different signs on the left and on the right of that point.

Tonio