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1 Further Integration
1.21
Example involving a quadratic
1.23
Complete example
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1.22
Example continued
Example
(Continued).
Evaluate
∫
0
2
4
x
+
1
x
2
+
4
d
x
.
\int_{0}^{2}\frac{4x+1}{x^{2}+4}\,dx.
By the previous slide, we have
∫
0
2
4
x
+
1
x
2
+
4
d
x
=
[
2
log
(
x
2
+
4
)
+
1
2
tan
-
1
x
2
]
0
2
\int_{0}^{2}\frac{4x+1}{x^{2}+4}\,dx=\left[2\log(x^{2}+4)+\frac{1}{2}\tan^{-1}% \frac{x}{2}\right]_{0}^{2}
=
2
log
8
+
1
2
tan
-
1
1
-
2
log
4
-
1
2
tan
-
1
0
=
2
log
2
+
π
8
.
=\,{2\log 8+\frac{1}{2}\tan^{-1}1-2\log 4-\frac{1}{2}\tan^{-1}0=}\,{2\log 2+% \frac{\pi}{8}.}