Trigonometric substitutions are used to solve certain types of integrals. We are taking the integral of
1
dx
x 2 +4
For x we can substitute 2tanu with dx= 2sec 2u du
Now we substitute our trigonometric function into
tan 1
x2 + 4
=>
4 tan2 u+ 4
=> 2secu

d erivatives are f ound by finding therate of change of on e variable with t he o ther v ariable r emaining
constant .
We will find
( 2 x 4 y +5 x y 2 )
x
4
We hold the variable y cons tan tuse the power rule find the derivative of x x .
4
3
The derivat

Fourier series are used to represent a periodic wave like function by integrating a series made of terms of
sines and cosines. A function can be represented by the sines and cosines in the form of:
f ( x )=a0+ a1 cosx +a2 cos 2 x+ +a n cos n x + b1 sin x

The transient heat conduction that used when tempering steel rods to determine
the hardness after quenching can be found by
x
2
= ex
0
2
where x is the diameter of rod
Now we can find the first and second derivatives
x
2
= ex
0
x
2
2
'=
2 x ex
0
2
x
2