Science:Math Exam Resources/Courses/MATH105/April 2012/Question 05 (a)/Solution 1

By the Fundamental Theorem of Calculus, if

$F(x)=\int _{0}^{x}\ln(2+\sin t)dt$

then

$\displaystyle F'(x)=\ln(2+\sin x)$.

Similarly,

$G(y)=\int _{y}^{0}\ln(2+\sin t)dt=-\int _{0}^{y}\ln(2+\sin t)dt$

so the Fundamental Theorem of Calculus tells us that

$\displaystyle G'(y)=-\ln(2+\sin y).$

We now can evaluate

$\displaystyle F'(\pi /2)=\ln(2+\sin(\pi /2))=\ln(2+1)=\ln 3$

and

$\displaystyle G'(\pi /2)=-\ln(2+\sin(\pi /2))=-\ln(2+1)=-\ln 3.$