Integration of Rational Functions, Resulting in Logarithmic or Arctangent Functions

THE INTEGRATION OF RATIONAL FUNCTIONS, RESULTING IN LOGARITHMIC OR ARCTANGENT FUNCTIONS

The following problems involve the integration of rational functions, resulting in logarithmic or inverse tangent functions. We will assume knowledge of the following well-known differentiation formulas :

$\displaystyle{ D \{ \ln \vert x\vert \} = { 1 \over x } }$ ,

where $\ln x$ represents the natural (base e) logarithm of x and

$\displaystyle{ D \{ \arctan x \} = { 1 \over 1+x^2 } }$ ,

where $\arctan x$ represents the inverse function of $\tan x$ . These formulas lead immediately to the following indefinite integrals :

$\displaystyle{ \int { 1 \over x } \,dx } = \ln \vert x\vert + C$ ,
$\displaystyle{ \int { 1 \over 1+x^2 } \,dx } = \arctan x + C$ .

Because the integral

$\displaystyle{ \int { 1 \over a^2+x^2 } \,dx }$ ,

where a is any positive constant, appears frequently in the following set of problems, we will find a formula for it now using u-substitution so that we don't have to do this simple process each time. Begin by rewriting the function. Then

$\displaystyle{ \int { 1 \over a^2+x^2 } \,dx } = \displaystyle{ \int { 1 \over a^2(1+(x^2/a^2)) } \,dx }$

$= \displaystyle{ { 1 \over a^2}\int { 1 \over 1+(x/a)^2 } \,dx }$ .

Now let

u= x/a

so that

du = (1/a) dx ,

(a)du = dx .

Now substitute into the original problem, replacing all forms of x, and getting

$\displaystyle{ { 1 \over a^2} \int { 1 \over 1+(x/a)^2 } \,dx } = \displaystyle{ { 1 \over a^2} \int { 1 \over 1+(u)^2 } \, (a)du }$

$= \displaystyle{ { a \over a^2} \int { 1 \over 1+u^2 } \, du }$

$= \displaystyle{ { 1 \over a} \int { 1 \over 1+u^2 } \, du }$

$= \displaystyle{ (1/a) \arctan u + C }$

$= \displaystyle{ (1/a) \arctan(x/a) + C }$ .

We now have the following variation of formula 1 :

3. $\displaystyle{ \int { 1 \over a^2+x^2 } \,dx } = \displaystyle{ (1/a) \arctan(x/a) + C }$

Recall also the method of algebraically "completing the square". For example to complete the square with x²+6x we divide 6 by 2, square it getting 9, add and subtract 9, so that

x²+6x = (x²+6x+9) - 9 = (x+3)² - 9 .

To complete the square with x²+Bx , divide B by 2, square it getting (B/2)²=B²/4, add and subtract B²/4, so that

x²+Bx = (x²+Bx+B²/4) - B²/4 = (x+(B/2))² - B²/4 .

As you do the following problems, remember these three general rules for integration :

$\displaystyle{ \int x^n \,dx } = { x^{n+1} \over n+1 } + C$ ,

where n is any constant not equal to -1,

$\displaystyle{ \int k f(x) \,dx } = k \displaystyle{ \int f(x) \,dx }$ ,

where k is any constant, and

$\displaystyle{ \int ( f(x) \pm g(x) ) \,dx } = \displaystyle{ \int f(x) \,dx } \pm \displaystyle{ \int g(x) \,dx }$ .

Most of the following problems are average. A few are challenging. Knowledge of the method of u-substitution will be required on many of the problems. Make precise use of the differential notation dx and du and always be careful when arithmetically and algebraically simplifying expressions. There is a very high emphasis on arithmetic and algebra in this problem set. AVOID MAKING THE FOLLOWING VERY COMMON ALGEBRA MISTAKE :

THIS IS FALSE : $\displaystyle{ A \over B+C } = \displaystyle{ { A \over B } + { A \over C } }$ .

Convince yourself that this is generally INCORRECT by plugging numbers in for A, B, and C (Try A=4, B=2, and C=2 .).

PROBLEM 1 : Integrate $\displaystyle{ \int { 3+x^2 \over x } \,dx }$ .
Click HERE to see a detailed solution to problem 1.
PROBLEM 2 : Integrate $\displaystyle{ \int { 7 \over x+5 } \,dx }$ .
Click HERE to see a detailed solution to problem 2.
PROBLEM 3 : Integrate $\displaystyle{ \int { 3 \over x^2+4 } \,dx }$ .
Click HERE to see a detailed solution to problem 3.
PROBLEM 4 : Integrate $\displaystyle{ \int { x \over x^2+4 } \,dx }$ .
Click HERE to see a detailed solution to problem 4.
PROBLEM 5 : Integrate $\displaystyle{ \int { x^3 \over x+2 } \,dx }$ .
Click HERE to see a detailed solution to problem 5.
PROBLEM 6 : Integrate $\displaystyle{ \int { x^4+x^3 \over x^2+1 } \,dx }$ .
Click HERE to see a detailed solution to problem 6.
PROBLEM 7 : Integrate $\displaystyle{ \int { 1 \over x (3+ \ln x ) } \,dx }$ .
Click HERE to see a detailed solution to problem 7.
PROBLEM 8 : Integrate $\displaystyle{ \int { 7 - \ln x \over x (3+ \ln x) } \,dx }$ .
Click HERE to see a detailed solution to problem 8.
PROBLEM 9 : Integrate $\displaystyle{ \int { 1 \over x^2+4x+5 } \,dx }$ .
Click HERE to see a detailed solution to problem 9.
PROBLEM 10 : Integrate $\displaystyle{ \int { 1 \over 2x^2+12x+68 } \,dx }$ .
Click HERE to see a detailed solution to problem 10.
PROBLEM 11 : Integrate $\displaystyle{ \int { 1 \over \sqrt{x} (x+9) } \,dx }$ .
Click HERE to see a detailed solution to problem 11.
PROBLEM 12 : Integrate $\displaystyle{ \int { \sin (2x) \over 1 + \cos (2x) } \,dx }$ .
Click HERE to see a detailed solution to problem 12.
PROBLEM 13 : Integrate $\displaystyle{ \int { \cos x \over 1 + \sin^2 x } \,dx }$ .
Click HERE to see a detailed solution to problem 13.
PROBLEM 14 : Integrate $\displaystyle{ \int { e^{2x} \over 1 + e^{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 14.
PROBLEM 15 : Integrate $\displaystyle{ \int { e^x \over 1 + e^{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 15.
PROBLEM 16 : Integrate $\displaystyle{ \int { e^{4x} \over 1 + e^{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 16.
PROBLEM 17 : Integrate $\displaystyle{ \int { x \over x^2+2x+1 } \,dx }$ .
Click HERE to see a detailed solution to problem 17.
PROBLEM 18 : Integrate $\displaystyle{ \int { x \over x^2+2x+5 } \,dx }$ .
Click HERE to see a detailed solution to problem 18.
PROBLEM 19 : Integrate $\displaystyle{ \int { 7x-2 \over 2x^2-16x+42 } \,dx }$ .
Click HERE to see a detailed solution to problem 19.
PROBLEM 20 : Integrate $\displaystyle{ \int { 1 \over 1 + e^{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 20.
PROBLEM 21 : Integrate $\displaystyle{ \int { e^{3x} \over 1 + e^{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 21.
PROBLEM 22 : Integrate $\displaystyle{ \int { 9+6 \sqrt{x}+x \over 4 \sqrt{x}+x } \,dx }$ .
Click HERE to see a detailed solution to problem 22.