# page 12

f<-function(x){sqrt(1+sin(x)^2/2)}
integrate(f,lower=0,upper=pi/2)

# page 18

f<-function(x){exp(sin(x))}
integrate(f,lower=0,upper=pi)

curve(f,xlim=c(-3*pi,3*pi),ylim=c(0,3))
curve(f,xlim=c(0,pi),ylim=c(0,3), type="h",lwd=0.2,add=T,col="magenta")
abline(h=0,lty=2)

# page 26

f<-function(x){x*exp(-x)}
integrate(f,lower=0,upper=Inf)

f<-function(x){x*exp(-x^2)}
integrate(f,lower=0,upper=Inf)
integrate(f,lower=-Inf,upper=Inf)

f<-function(x){1/sqrt(1-x^2)}
integrate(f,lower=0,upper=1)
pi/2

f<-function(x){x/(1+x^2)}
integrate(f,lower=0,upper=Inf)     # indeterminate form ∞
integrate(f,lower=-Inf,upper=Inf)  # don't do this, it is not correct here

f<-function(x){1/(x*sqrt(x^2-1))}
integrate(f,lower=sqrt(2),upper=Inf)     
pi/4