Here's the question you clicked on:
abp2u
Integration: (-pi/4) to (pi/4) cos x dx
Do you know the Fundamental Theorem of Calculus?
note that \(\int cos x dx= -sin x +C\)
It says\[\int\limits_{a}^{b}f(x)dx=F(a)-F(b)\]In this case this means: (see also the answer of Shadowys)\[\int\limits_{-\frac{ \pi }{ 4 }}^{\frac{ \pi }{ 4 }}cosxdx=\sin(\frac{ \pi }{ 4 })-\sin( -\frac{ \pi }{ 4 })=\frac{ 1 }{ 2 }\sqrt{2}--\frac{ 1 }{ 2 }\sqrt{2}=\sqrt{2}\]
F is called a primitive function of f. It means: F'(x) = f(x). So the Fundamental Theorem makes integrating (= calculating an infinite sum of infinite small numbers - very hard!) much easier: if you can find a primitive F of f, you're done. In the case of cos(x) this is simple: (sinx)' = cos x, so F(x) = sinx.