Math Help - definite integrals

1. definite integrals

1. how do I show that cos(x^2) is greater than or equal to cos(x) for 0 < x < 1?

2. deduce that integral of cos(x^2)dx from 0 to pi/6 is greater than or equal to 1/2.

2. Re: definite integrals

Originally Posted by Taurus3
1. how do I show that cos(x^2) is greater than or equal to cos(x) for 0 < x < 1?

2. deduce that integral of cos(x^2)dx from 0 to pi/6 is greater than or equal to 1/2.
1) The $\cos$ is a decreasing function on $(0,1)~.$
If $x\in (0,1)$ then $x^2 so $\cos(x^2)>\cos(x)~.$

2) There is a theorem: If each of $f~\&~g$ is integrable and $\left( {\forall x \in [a,b]} \right)\left[ {f(x) \leqslant g(x)} \right]$ then $\int_a^b f \leqslant \int_a^b g$.