Calculus UNIT509: What is an Indefinite Integral?

2770 days ago by MATH4R2013

#1) 88AB6: f'(x)=a*x^2+b*x, f'(1)=6, f"(1)=18, integrate(f(x),x,1,2)=18, find f(x)! var('a,b') fp(x)=a*x^2+b*x fpp(x)=diff(fp(x),x) show(fp(x)) show(fpp(x)) 
       

show(fp(1)) show(fpp(1)) 
       

solve([6==a+b,18==2*a+b],(a,b)) 
       
fp(x)=12*x^2-6*x f(x)=integrate(fp(x),x) show(fp(x)) show(f(x)) 
       

var('C') f(x)=4*x^3-3*x^2+C show(integrate(f(x),x)) show(integrate(f(x),x,1,2)) 
       

solve(C+8==18,C) 
       
f(x)=4*x^3-3*x^2+10 show(f(x)) 
       
#2) T/F: integrate(cos(x)^2,x)==integrate(1/2+cos(2*x)/2,x), (True) show(integrate(cos(x)^2,x)) show(integrate(1/2+cos(2*x)/2,x)) 
       

#3) T/F: integrate(sin(x)^2,x)==integrate(1/2-cos(2*x)/2,x), (True) show(integrate(sin(x)^2,x)) show(integrate(1/2-cos(2*x)/2,x)) 
       

#4) T/F: integrate(x*cos(x),x)==x*sin(x)+C, (False) show(diff(x*sin(x),x)) 
       
#5) T/F: integrate(x*cos(x),x)==x*cos(x)+C, (False) show(diff(x*cos(x),x)) 
       
#6) T/F: integrate(x*cos(x),x)==x*sin(x)-cos(x)+C, (False) show(diff(x*sin(x)-cos(x),x)) 
       
#7) T/F: integrate(x*cos(x),x)==x*sin(x)+cos(x)+C, (True) show(diff(x*sin(x)+cos(x),x))