http://www.ck12.org Chapter 7. Integration Techniques
since
cotθ=cossinθθ.so that
∫ dx
x^2 √ 4 −x^2 =−1
4 cotθ+C
=−^14√
4 −x^2
x +C.Example 2:
Evaluate∫
√x (^2) − 3
x dx.
Solution:
Again, we want to first to eliminate the radical. Consult the table above and substitutex=
√
√ 3 secθ. Thendx=
3 secθtanθdθ. Substituting back into the integral,
∫ √x (^2) − 3
x dx=
∫ √3 sec (^2) θ− 3
√3 secθ
√
3 secθtanθdθ
=√
3
∫
tan^2 θdθ.Using the integral identity from the section on Trigonometric Integrals,
∫
tanmxdx=tanm− (^1) x
m− 1 −
∫
tanm−^2 xdx.
and lettingm=2 we obtain
∫ √x (^2) − 3
x dx=
√ 3 ((tanθ)−θ)+C.
Looking at the triangles above, the third triangle represents our case, witha=
√
- Sox=
√
3 secθand thus
cosx=
√
3 /x, which gives tanθ=√
x^2 − 3 /√
- Substituting,
∫ √x (^2) − 3
x dx=
√ 3 ((tanθ)−θ)+C
=√ 3
(√
x√^2 − 3
3 −tan− 1(√
x√^2 − 3
3))
+C
=
√
x^2 − 3 −√3 tan−^1(√
x√^2 − 3
3