728
Chapter 6 | Applications of Integration
d dx
e x = e x
as desired, which leads immediately to the integration formula
∫ e x dx = e x + C .
We apply these formulas in the following examples.
Example 6.38 Using Properties of Exponential Functions
Evaluate the following derivatives: a. d dt e 3 t e t 2
2
d dx
e 3 x
b.
Solution We apply the chain rule as necessary. a. d dt e 3 t e t 2 = d dt e 3 t + t 2 = e 3 t + t 2
(3+2 t )
2
2
d dx
e 3 x
= e 3 x
6 x
b.
6.38
Evaluate the following derivatives:
⎛ ⎝ ⎜ e x
⎞ ⎠ ⎟
2 e 5 x
d dx
a.
3
⎛ ⎝ e 2 t
⎞ ⎠
b. d dt
Example 6.39 Using Properties of Exponential Functions Evaluate the following integral: ∫ 2 xe − x 2 dx .
Solution Using u -substitution, let u =− x 2 . Then du =−2 xdx , and we have ∫ 2 xe − x 2 dx =− ∫ e u du =− e u + C =− e − x 2 + C .
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online