02-NOV

Ideas to find anti-derivatives

1. Know the table of basic integrals.

2. identify the given function, is it algebraic (monomial/power functions)? trigonometric? expoential? logarithmic? or anti-trigonometric?

3. Or the sum of these five types? Then separate the summands.(see thm. 4.16)

4. Or is it a product of these five types? Then integral by parts (—> the derivative formula for products)

5. Try substitution, to reduce to the above cases.

6. The case of rational functions (reduction to partial fractions)

Glossary Remark: anti-derivative = indefinite integral

Example 4.50

a. Which ) will have derivative %3D3x%5E2#card=math&code=F%27%28x%29%3D3x%5E2)? %20%3D%20x%5En%20%5CRightarrow%20F’(x)%3Dn%20x%5E%7Bn-1%7D#card=math&code=F%28x%29%20%3D%20x%5En%20%5CRightarrow%20F%27%28x%29%3Dn%20x%5E%7Bn-1%7D). Comparing the exponent, we have . So . So that %3Dx%5E3#card=math&code=F%28x%29%3Dx%5E3) gives %3D3x%5E2#card=math&code=F%27%28x%29%3D3x%5E2). Thus, all anti-derivatives for are

b. . So all anti-derivatives are %2BC#card=math&code=%5Cln%28x%29%2BC)

c. ‘%3D%5Ccos(x)#card=math&code=%5Csin%28x%29%27%3D%5Ccos%28x%29) So all anti-derivatives are %2BC#card=math&code=%5Csin%28x%29%2BC)

d. ‘%20%3D%20e%5Ex#card=math&code=%28e%5Ex%29%27%20%3D%20e%5Ex). So all anti-derivatives are .

Comparison of derivatives and anti-derivatives:

1. About summation. ‘%3D%20f’%2Bg’#card=math&code=%28f%2Bg%29%27%3D%20f%27%2Bg%27), %2Bg(x))%20dx%20%3D%20%5Cint%20f(x)dx%20%2B%20%5Cint%20g(x)%20dx#card=math&code=%5Cint%20%28f%28x%29%2Bg%28x%29%29%20dx%20%3D%20%5Cint%20f%28x%29dx%20%2B%20%5Cint%20g%28x%29%20dx)

2. about constant multiple: )’%20%3D%20C%20f’(x)#card=math&code=%28Cf%28x%29%29%27%20%3D%20C%20f%27%28x%29) , )%20dx%20%3D%20C%20%5Cint%20f(x)%20dx#card=math&code=%5Cint%20%28C%20f%28x%29%29%20dx%20%3D%20C%20%5Cint%20f%28x%29%20dx) .
   The above two properties are summarized in thm 4.16.

3. about scalar , the power rule for integrals (thm 4.15 in the textbook) only requires that .
   Now we take in thm 4.15, we get (because )

The indefinite integral for exponential functions

$\int a^x dx = \frac{a^x}{\ln a} +C $ <—> ‘%3Da%5Ex%20%5Cln%20a#card=math&code=%28a%5Ex%29%27%3Da%5Ex%20%5Cln%20a)

Verification: ‘%20%3D%20%5Cfrac%7B1%7D%7B%5Cln%20a%7D%20(a%5Ex)’%20%3D%20%5Cfrac%7B1%7D%7B%5Cln%20a%7D%20a%5Ex%20%5Cln%20a%20%3Da%5Ex#card=math&code=%28a%5Ex%2F%5Cln%20a%29%27%20%3D%20%5Cfrac%7B1%7D%7B%5Cln%20a%7D%20%28a%5Ex%29%27%20%3D%20%5Cfrac%7B1%7D%7B%5Cln%20a%7D%20a%5Ex%20%5Cln%20a%20%3Da%5Ex)

Now if , then . So

%2Bg(x))%20dx%20%3D%20%5Cint%20f(x)%20dx%20%2B%20%5Cint%20g(x)dx%20%20%3DF(x)%20%2BC_1%20%2BG(x)%20%2BC_2%20%3DF(x)%2BG(x)%20%2BC#card=math&code=%5Cint%20%28f%28x%29%2Bg%28x%29%29%20dx%20%3D%20%5Cint%20f%28x%29%20dx%20%2B%20%5Cint%20g%28x%29dx%20%20%3DF%28x%29%20%2BC_1%20%2BG%28x%29%20%2BC_2%20%3DF%28x%29%2BG%28x%29%20%2BC)

Only one constant at the last step is enough.

Example 4.52

$x^{1/3}/x = x{-2/3} $ Now use the power rule %20%2BC#card=math&code=%5Cint%20x%5En%20dx%20%3D%20x%5E%7Bn%2B1%7D%2F%28n%2B1%29%20%2BC)

%5Ccos(x)%20%3D%20%5Csin(x)#card=math&code=%5Ctan%28x%29%5Ccos%28x%29%20%3D%20%5Csin%28x%29), %20dx%20%3D%20-%5Ccos(x)%2BC#card=math&code=%5Cint%20%5Csin%28x%29%20dx%20%3D%20-%5Ccos%28x%29%2BC)

Prob. 472. %20x%5E2%20-%20x%5E3%2F3%20%2BC#card=math&code=%5Cint%20e%5Ex%20%2B%203x%20-x%5E2%20dx%20%3D%20%5Cint%20e%5Ex%20dx%20%2B%203%20%5Cint%20x%20dx%20-%20%5Cint%20x%5E2%20dx%20%3D%20%20%20%20e%5Ex%20%20%2B%20%283%2F2%29%20x%5E2%20-%20x%5E3%2F3%20%2BC)

Prob. 498 To find out , we shall ask which function #card=math&code=F%28x%29) will have %3De%5E%7B-x%7D#card=math&code=F%27%28x%29%3De%5E%7B-x%7D) .

‘%20%3D%20-%20e%5E%7B-x%7D#card=math&code=%28e%5E%7B-x%7D%29%27%20%3D%20-%20e%5E%7B-x%7D) . So %20%3D%20-e%5E%7B-x%7D#card=math&code=F%28x%29%20%3D%20-e%5E%7B-x%7D) This implies that