I am not supposed to "integrate by substitution" or anything fancy yet in this section ......

"In each part, confirm that the formula is correct and state a corresponding integration formula."

a)(d/dx)[sqrt(1+ X^{2})] = (X)(1+X^{2})^{-1/2 }

Integral (X)(1+X^{2})^{-1/2 }dx

I understand how to take the integral of (1+X^{2})^{-1/2 }, but where does the x go? is it x(dx) and becomes 1? and then whats leftover is the integral of (X)(1+X^{2})^{-1/2 }??? That would make sense, I just figured that out while I was typing it but I wanted to make sure ...

b)(d/dx)[1/3 sin(1+x^{3})] = x^{2}cos(1+x^{3})

Integral x^{2}cos(1+x^{3})dx

I got (x^{3 }/3)(-sin(1+x^{3})(3x^{2}) ... I have no idea how that turns out to be what it is....

c)"Find the derivative and state a corresponding integration formula."

(d/dx)[sqrt(x^{3}+5)] = (3x^{2})/(2)[sqrt(x^{3}+5)]

Integral (3x^{2})/(2)[sqrt(x^{3}+5)] dx

Once again... I understand how to take the integral of [sqrt(x^{3}+5)]_{-1/2}.... but I don't understand what happens to (3/2)x^{2 }...?

Thank you!