Chapter 6 | Applications of Integration
751
Solution Using the formulas in Table 6.4 and the chain rule, we obtain the following results: a. d dx ⎛ ⎝ sinh −1 ⎛ ⎝ x 3 ⎞ ⎠ ⎞ ⎠ = 1 2 = 1
9+ x 2
3 1+ x
9
⎛ ⎝ tanh −1 x ⎞ ⎠
2
2
⎛ ⎝ tanh −1 x ⎞ ⎠
d dx
=
b.
1− x 2
6.49
Evaluate the following derivatives:
⎛ ⎝ cosh −1 (3 x ) ⎞ ⎠
d dx d dx
a.
3
⎛ ⎝ coth −1 x ⎞ ⎠
b.
Example 6.50 Integrals Involving Inverse Hyperbolic Functions
Evaluate the following integrals: a. ∫ 1 4 x 2 −1 dx b. ∫ 1 2 x 1−9 x 2 dx
Solution We can use u -substitution in both cases. a. Let u =2 x . Then, du =2 dx and we have ∫ 1 4 x 2 −1 dx = ∫ 1 2 u 2 −1 b. Let u =3 x . Then, du =3 dx and we obtain ∫ 1 2 x 1−9 x 2 dx = 1 2 ∫ 1 u 1− u 2
−1 u + C = 1
−1 (2 x )+ C .
du = 1
2 cosh
2 cosh
−1 | u | + C = − 1
−1 |3 x | + C .
du = − 1
2 sech
2 sech
6.50
Evaluate the following integrals: a. ∫ 1 x 2 −4 dx , x >2 b. ∫ 1 1− e 2 x dx
Made with FlippingBook - professional solution for displaying marketing and sales documents online