http://www.ck12.org Chapter 6. Transcendental Functions
The Integration Formulas of the Inverse Trigonometric Functions
The derivative formulas in the box above yield the following integrations formulas for inverse trigonometric func-
tions:
∫ du
√ 1 −u 2 =sin−^1 u+c
∫ du
1 +u^2 =tan− (^1) u+c
∫ du
u√u^2 − 1 =sec
− 1 ∣∣u∣∣+c
Example 6:
Evaluate∫ 1 +dx 4 x 2.
Solution:
Before we integrate, we useu−substitution. Letu= 2 x(the square root of 4x^2 ). Thendu= 2 dx.Substituting,
∫ dx
1 + 4 x^2 =
∫ 1 / 2
1 +u^2 du
=^12∫ 1
1 +u^2 du
=^12 tan−^1 u+c
=^12 tan−^1 ( 2 x)+c.Example 7:
Evaluate∫√ 1 e−xe 2 xdx.
Solution:
We useu−substitution. Letu=ex,sodu=exdx.Substituting,
∫ ex
√ 1 −e 2 xdx=∫ ex
√ 1 −u 2 duex=∫ 1
√ 1 −u 2 du
=sin−^1 u+c
=sin−^1 (ex)+c.Example 8:
Evaluate the definite integral∫ln(^2 /
√ 3 )
ln2 e
√ −x
1 −e−^2 xdx.
Solution:
Substitutingu=e−x,du=−e−xdx.