integration questions

i just read some of examworld's integration questions and dont really get some of the steps in the answers.

(1) Integrate [(x + 1)³ + 1](x + 1)²

um where did the 1/3 and the 3 come from?

and (5) Integrate (1 - sin2x)/(x - sin² x)

someone mind explaining?

y = (x + 1)^3

dy/dx = 3(x + 1)^2

The 3(x + 1)^2 is the derivative of (x + 1)^3, you just need to prefix a "3" in front, so to cancel out the 3, you do a 1/3.

Differentiation product rule, chain rule also can :

u = x - (sin x)^2

u' = 1 - 2 sin x cos x

u' = 1 - sin 2x ===>>> (sin 2x = 2 sin x cos x) aka double angle formula

Integrate [1/u] du = ln u + c (I cannot do the integral sign)

Then all you have to do is slot in "u" into "ln u + c":

===>>> ln (x - (sin x)^2) + c

I just learnt this just now.

thanks for helping to explain

Am glad something new is learned :D

Just realised that there is something wrong with your answer for question 2.

It should be ln |x - (sin x)^2| + c

It should be the absolute value.