MATH—166
Spring 2012
Chapter 7 - Summary
7.1 Basic Integration Rules
Following are some of the important Integral forms to remember. See page 383—384 in the textbook for
more Integral forms.
 r+1

Z
r 6= −1
 ur+1 + C
r
1.
x dx =

 ln |x| + C
r = −1
Z
x
dx
√
2.
+C
= sin−1
a
a 2 − x2
Z
dx
1
−1 x
3.
+C
=
tan
a2 + x2
a
a
Z
dx
1
|x|
1
a
−1
−1
√
4.
= sec
+ C = cos
+C
2
2
a
a
a
|x|
x x −a
5. Let g be a differentiable function and suppose that F is an anti-derivative of f . Then, if u = g(x),
Z
Z
0
f g(x) g (x)dx =
f (u)du = F (u) + C = F g(x) + C
7.2 Integration by Parts
Integration by Parts Formula
Z
Z
udv = uv − vdu
7.3 Some Trigonometric Integrals
Useful Identities
Half-Angle Identities
sin2 x + cos2 x = 1
sin2 x =
1 − cos 2x
2
1 + tan2 x = sec2 x
cos2 x =
1 + cos 2x
2
1 + cot2 x = csc2 x
Z
1.
n
sin x dx
Z
n
cos x dx
(a) When n is an odd positive integer.
First, factor out a sin x (or a cos x). Then use the identity sin2 x + cos2 x = 1 to replace the
remaining (n − 1) even sin x (or cos x) terms.
(b) When n is an even positive integer.
Use half-angle identities to replace every sin2 x (or cos2 x).
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MATH—166
Z
Spring 2012
sinm x cosn x dx
2.
(a) When m or n odd.
For the odd component, use strategy 1.(a) above.
(b) For every even component, use strategy 1.(b) above.
.
Z
Z
Z
3.
sin mx cos nx dx,
sin mx sin nx dx,
cos mx cos nx dx
Use the following Product Identities:
1
sin mx cos nx = [sin(m + n)x + sin(m − n)x]
2
1
sin mx sin nx = − [cos(m + n)x − cos(m − n)x]
2
1
[cos(m + n)x + cos(m − n)x]
2
cos mx cos nx =
Z
tann x dx,
4.
Z
cotn x dx
• In the Tangent case, factor out tan2 x and replace with tan2 x = sec2 x − 1.
• In the Cotangent case, factor out cot2 x and replace with cot2 x = csc2 x − 1.
Z
tanm x secn x dx,
5.
Z
cotm x cscn x dx
• n Even, m Any Number
Use strategy 4. for the even component.
• n Odd, m Any Number
Use strategy 4. for (n − 1) even components.
7.4 Rationalizing Substitutions
√
n
ax + b appears in the integral:
√
n
Substitute u = ax + b.
1. If
2. If
√
a2 − x2 ,
Radical
√
a 2 − x2
√
a 2 + x2
√
x2 − a2
√
a2 + x2 or
√
x2 − a2 appears in the integral:
Substitution
Restriction on t
x = a sin t
− π2 ≤ t ≤
π
2
x = a tan t
− π2 ≤ t ≤
π
2
x = a sec t
0 ≤ t ≤ π, t 6=
π
2
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MATH—166
Spring 2012
Note: Use the above trigonometric substitutions only if the Integrand is not of the form 7.1 #5. OR
you cannot convert the Integrand to the form 7.1 #5 by Scalar Multiplication/Division and/or
scalar Addition/Subtraction.
3. If a quadratic expression of the form ax2 + bx + c appears under the radical, complete the square and
use an appropriate substitution.
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