Example. Evaluate
�
1
𝑑𝑑𝑥𝑥
1 + cos 𝑥𝑥
Example. Evaluate
�
1
𝑑𝑑𝑥𝑥
𝑒𝑒 𝑥𝑥 + 𝑒𝑒 −𝑥𝑥
Integration of Rational Functions
When integrating rational functions (polynomials in the numerator and
denominator), check to see if the degree of the numerator is greater than or equal
to the degree of the denominator. If so, a long division is used to simplify it.
Example. Evaluate
�
1
𝑑𝑑𝑥𝑥.
𝑥𝑥 + 1
Example. Evaluate
𝑥𝑥 + 2
�
𝑑𝑑𝑥𝑥
𝑥𝑥 + 1
Example. Evaluate
�
1
𝑑𝑑𝑥𝑥
𝑥𝑥 2 + 1
Example. Evaluate
�
𝑥𝑥
𝑑𝑑𝑥𝑥
𝑥𝑥 2 + 1
Example. Evaluate
𝑥𝑥 2
� 2
𝑑𝑑𝑥𝑥
𝑥𝑥 + 1
Example. Evaluate
�
1 − 𝑥𝑥
𝑑𝑑𝑥𝑥
1 + 𝑥𝑥 2
Integration by Completing the Squa
re
To complete the square of the quad
ratic polynomial
𝑥𝑥 2 + 𝑥𝑥 +
we follow the following steps:
𝑥𝑥 2
+ 𝑥𝑥 +
= 𝑥𝑥 2
2
+ 𝑥𝑥 +
2
2
−
Example (An Integral Where We Must Co
�
1
√−5 + 6𝑥𝑥 − 𝑥𝑥 2
𝑑𝑑𝑥𝑥
2
2
+ = 𝑥𝑥 +
2
2
−
4
+
mplete the Square). Evaluate
Example (An Integral Requiring Some Imagination). Evaluate
�
4𝑥𝑥 + 1
2𝑥𝑥 2 + 4𝑥𝑥 + 10
𝑑𝑑𝑥𝑥