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The integration-by-parts formula (Equation \ref{IBP}) allows the exchange of one integral for another, possibly easier, integral. Integration by parts applies to both definite and indefinite integrals.
- Exercises for Section 7.1
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- Exercises for Section 7.1
Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.
What is integration by parts? Integration by parts is a method to find integrals of products: ∫ u ( x) v ′ ( x) d x = u ( x) v ( x) − ∫ u ′ ( x) v ( x) d x. or more compactly: ∫ u d v = u v − ∫ v d u.
Integrals of Exponential and Logarithmic Functions. ∫ ln x dx = x ln x − x + C. + 1 x. + 1. x ∫ x ln xdx = ln x − + C. 2 + 1 ( n + 1 ) x dx = e x + C ∫.
Integrals with Trigonometric Functions Z sinaxdx= 1 a cosax (63) Z sin2 axdx= x 2 sin2ax 4a (64) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (65) Z sin3 axdx= 3cosax 4a + cos3ax 12a (66) Z cosaxdx=
Indefinite Integrals Rules: ∫Integration By Parts: ′= −∫ ′ ∫Integral of a Constant: ( ) 𝑥=𝑥⋅ ( ) ∫Taking a Constant out: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫Sum/Difference Rule: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥
TABLE OF INTEGRALS. Note: the use of "a", "b" and "c" are as constants. xn. +. 1. ∫. xndx. = n. +. 1. 2. ∫. dx. = ln. x. 3. ∫. 1. = 1. dx. ln( ax. +. c. ) ax. +. c. a. 4. ∫. e x dx. = ex. 5. ∫. ( ax.
