Integral Calculus Answers

What is the definite integral of this |sin x| . limit is from -pi/2 to +pi/2?

i want to integrate this x*sqrt (a^2 - x^2)
let x= a sin z ; differentiating both side with respect to x , it becomes
1= a cos z * dz/dx
or dx = a cos z dz


if i differentiate both side with respect to z , it becomes

dx/dz =acosz
or dx = ac cos z dz

my question is which one is correct? differentiating with respect of x or with respect of z and why ?

what is the integration of 1/[{sinx}^4 + {cosx}^4]

what is the integration of 1/sqrt of [(x-1)(x-2)(x-3)] dx

Integrate with respect to v :
∫104v−632v2dv
8
58
−85
85

Integrate with respect to x :
∫1−36x2−5x+2dx
84
48
-84
-48
Integrate with respect to v :
∫104v−632v2dv
8
58
−85
85

∫1 / [(x)^m +(x)^n] dx =? show process please.? where m and n are integers

∫1 / [(x)^3 +(x)^8] dx =? show process please.?

∫ {(x)^4} / [(x)^3 + (x)^(-2)] dx =? show process please.?