618
Chapter 5 | Integration
If f ( x ) is continuous over an interval ⎡ ⎣ a , b ⎤
⎦ , then there is at least one point c ∈ ⎡ ⎣ a , b ⎤
⎦ such that
b − a ∫ a b
f ( c ) = 1
f ( x ) dx .
• Fundamental Theorem of Calculus Part 1 If f ( x ) is continuous over an interval ⎡ ⎣ a , b ⎤ F ′( x ) = f ( x ). • Fundamental Theorem of Calculus Part 2 If f is continuous over the interval ⎡ ⎣ a , b ⎤
⎦ , and the function F ( x ) is defined by F ( x ) = ∫ a x
f ( t ) dt , then
⎦ and F ( x ) is any antiderivative of f ( x ), then ∫ a b
f ( x ) dx = F ( b )− F ( a ).
• Net Change Theorem F ( b ) = F ( a )+ ∫ a b
F '( x ) dx or ∫ a b
F '( x ) dx = F ( b )− F ( a )
• Substitution with Indefinite Integrals ∫ f ⎡ ⎣ g ( x ) ⎤
⎦ g ′( x ) dx = ∫ f ( u ) du = F ( u )+ C = F ⎛
⎞ ⎠ + C
⎝ g ( x )
• Substitution with Definite Integrals ⌠ ⌡ a b f ⎛ ⎝ g ( x ) ⎞ ⎠ g '( x ) dx = ∫ g ( a ) g ( b ) f ( u ) du • Integrals of Exponential Functions ∫ e x dx = e x + C ⌠ ⌡ a x dx = a x ln a + C • Integration Formulas Involving Logarithmic Functions ∫ x −1 dx = ln| x | + C ∫ ln xdx = x ln x − x + C = x (ln x −1)+ C ∫ log a xdx = x ln a (ln x −1)+ C • Integrals That Produce Inverse Trigonometric Functions
⌠ ⌡ ⌠ ⌡ ⌠ ⌡
= sin −1 ⎛
⎞ ⎠ + C
du a 2 − u 2
⎝ u a
−1 ⎛
⎞ ⎠ + C
du a 2 + u 2
⎝ u a
= 1 a tan
−1 ⎛
⎞ ⎠ + C
du u u 2 − a 2
⎝ u a
= 1 a sec
KEY CONCEPTS 5.1 Approximating Areas • The use of sigma (summation) notation of the form ∑ i =1 n
a i is useful for expressing long sums of values in compact
form.
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