Math 1B
§ 7.5 Strategy for Integration
The challenge in this section is to recognize which technique or formula to use.
First, you must know the basic integration formulas (see the table on page 495). Then, try this four-step
strategy.
1. Simplify the integrand if possible – algebraic manipulation or trig identities.
2. Look for a u-substitution.
3. Classify the integrand according to its form.
a) Trig functions – powers of sinx & cosx or tanx & secx or cotx & cscx. Use the
substitutions from 7.2.
b) Rational functions – use partial fractions.
c) Integration by parts – product of a power of x (polynomial) and a transcendental
function (trig, exponential, logarithmic), or inverse trig function.
(
d) Radicals – trig sub or use a rationalizing substitution u = n g ( x )
4. Try again.
Example: Find
Stewart – 7e
∫
)
sin 3 x
dx
cos x
1
Example: Find
Example: Find
∫ te
2
2
∫
0
Stewart – 7e
t
dt
x2
1− x 2
dx
2
Example: Find
∫
Example: Find
∫
Stewart – 7e
1+ e x dx
ln x
x 1+ ( ln x )
2
dx
3
Example: Find
5
3
∫ tan θ sec θ dθ
€
Stewart – 7e
4