Basic Integration Rules
In this section, we will study several integration techniques for fitting an integrand into one of the basic integration rules. The basic integration rules are reviewed in Table 6.1 on page 252 .
Procedures for Fitting Integrands to Basic Rules
Procedure Example
1. Separating numerator $\frac{1-2 x}{1+x^2}=\frac{1}{1+x^2}-\frac{2 x}{1+x^2}$
2. Adding and subtracting terms in numerator $\frac{1}{1-e^x}=\frac{1-e^x+e^x}{1-e^x}=\frac{1-e^x}{1-e^x}+\frac{e^x}{1-e^x}$
3. Dividing improper fractions $\frac{x^3-3 x}{x^2-1}=x-\frac{2 x}{x^2-1}$
4. Completing the square $\frac{1}{\sqrt{4 x-x^2}}=\frac{1}{\sqrt{4-(x-2)^2}}$
Other integration techniques, such as the simple substitution method, were covered in section 4.8. Using trigonometric identities, trigonometric substitution, Method of Partial Fractions and Integration by Parts will be covered later in this chapter.
1. Separating numerator
Example 1
- Evaluate $\int \frac{1-2 x}{1+x^2} d x$.
▶️Answer/Explanation
Solution
$\int \frac{1-2 x}{1+x^2} d x=\int \frac{1}{1+x^2} d x+\int \frac{-2 x}{1+x^2} d x$ Separate the numerator
$\int \frac{1}{1+x^2} d x=\arctan x$ Basic integration rules
$\int \frac{-2 x}{1+x^2} d x=\int \frac{-d u}{u}=-\ln u$ Let $u=1+x^2$, then $d u=2 x d x$.
Therefore $\int \frac{1-2 x}{1+x^2} d x=\arctan x-\ln \left(1+x^2\right)+C$.
Differentiation Rules and Basic Integration Rules
Adding and subtracting terms in numerator
Example 2
- Evaluate $\int \frac{1}{1-e^x} d x$.
▶️Answer/Explanation
Solution
$\int \frac{1}{1-e^x} d x=\int \frac{1-e^x+e^x}{1-e^x} d x$ Add and subtract $e^x$ in the numerator.
$=\int \frac{1-e^x}{1-e^x} d x+\int \frac{e^x}{1-e^x} d x$ Separate the numerator
$=\int d x+\int \frac{e^x}{1-e^x} d x$
$=x-\ln \left(1-e^x\right)+C$ Use the basic integration rules.
Dividing improper fractions
Example 3
- Evaluate $\int \frac{x^3-3 x}{x^2-1} d x$.
▶️Answer/Explanation
Solution
$
\int \frac{x^3-3 x}{x^2-1} d x=\int\left(x-\frac{2 x}{x^2-1}\right) d x
$ Divide an improper fraction.
$=\int x d x-\int \frac{2 x}{x^2-1} d x$
$=\frac{1}{2} x^2-\ln \left(x^2-1\right)+C $ Use the basic integration rules.
Example4
- $
\int \frac{1+\sin x}{\cos ^2 x} d x=
$
(A) $\tan x-\sec x \tan x+C$
(B) $\tan x+\sec x+C$
(C) $\tan x+\sec ^2 x+C$
(D) $\ln \left(1+\cos ^2 x\right)+C$
▶️Answer/Explanation
Ans:B
Example5
- $
\int 2 \tan x \ln (\cos x) d x=
$
(A) $\cos x[\ln (\cos x)]+C$
(B) $\sin x[\ln (\cos x)]+C$
(C) $-[\ln (\cos x)]^2+C$
(D) $[\ln (\sin x)]^2+C$
▶️Answer/Explanation
Ans:D
Example6
- $
\int 2 \tan x \ln (\cos x) d x=
$
(A) $\cos x[\ln (\cos x)]+C$
(B) $\sin x[\ln (\cos x)]+C$
(C) $-[\ln (\cos x)]^2+C$
(D) $[\ln (\sin x)]^2+C$
▶️Answer/Explanation
Ans:C