8.1 Basic Integration Rules
In this section we will use various strategies to rewrite the expression to be integrated into a form that is more
convenient for integration.
Strategies:
I.
Separate the numerator.
!
II.
x+3
4 " x2
(Hint: Rewrite the fraction as the sum of two fractions.)
dx
Add and subtract terms in the numerator.
1
!1+ e
x
dx
(Hint: Add and subtract e x in the numerator and then
rewrite as the sum of two fractions.)
Scherer
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III.
Complete the square.
!
1
2x " x 2
dx
(Hint: When you complete the square in the denominator,
you may end up with an integral for arc sin, arc tan, or
arc sec.)
IV.
Divide a Rational Function.
x2
! x 2 + 1 dx
Scherer
(Hint: If the degree of the numerator is greater than or
equal to the degree of the denominator, it may be helpful to
divide the numerator by the denominator.)
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V.
Expand the expression.
! (1 + e
VI.
) dx
(Hint: Square the binomial before integrating.)
Use Trigonometric Identities.
! cot
Scherer
x 2
2
xdx
(Hint: Use a trig identity to rewrite the integrand. In this
case, use a Pythagorean Identity.)
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VII.
Multiply and Divide by the Conjugate.
1
! 1 + sin x dx
Scherer
(Hint: Multiply numerator and denominator by (1 – sin x)
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