Integrals of Functions Involving Absolute Values In Exercises 45–52, use integration by parts to evaluate the given integral using the following integral formulas where necessary. (You have seen some of these before; all can be checked by differentiating.) Integral Formula Shortcut Version ∫ | x | x d x = | x | + C Because d d x | x | = | x | x ∫ | a x + b | a x + b d x = 1 a | a x + b | + C ∫ | x | d x = 1 2 x | x | + C ∫ | a x + b | d x = 1 2 a ( a x + b ) | a x + b | + C ∫ x | x | d x = 1 3 x 2 | x | + C ∫ ( a x + b ) | a x + b | d x = 1 3 a ( a x + b ) 2 | a x + b | + C ∫ x 2 | x | d x = 1 4 x 3 | x | + C ∫ ( a x + b ) 2 | a x + b | d x = 1 4 a ( a x + b ) 3 | a x + b | + C ∫ 3 x | x + 4 | x + 4 d x
Integrals of Functions Involving Absolute Values In Exercises 45–52, use integration by parts to evaluate the given integral using the following integral formulas where necessary. (You have seen some of these before; all can be checked by differentiating.) Integral Formula Shortcut Version ∫ | x | x d x = | x | + C Because d d x | x | = | x | x ∫ | a x + b | a x + b d x = 1 a | a x + b | + C ∫ | x | d x = 1 2 x | x | + C ∫ | a x + b | d x = 1 2 a ( a x + b ) | a x + b | + C ∫ x | x | d x = 1 3 x 2 | x | + C ∫ ( a x + b ) | a x + b | d x = 1 3 a ( a x + b ) 2 | a x + b | + C ∫ x 2 | x | d x = 1 4 x 3 | x | + C ∫ ( a x + b ) 2 | a x + b | d x = 1 4 a ( a x + b ) 3 | a x + b | + C ∫ 3 x | x + 4 | x + 4 d x
Solution Summary: The author explains how to calculate the value of integral displaystyleint 3xleft|x+4right|x + 4dx
Integrals of Functions Involving Absolute Values In Exercises 45–52, use integration by parts to evaluate the given integral using the following integral formulas where necessary. (You have seen some of these before; all can be checked by differentiating.)
Integral Formula
Shortcut Version
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With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Graphing Inverse Functions
Each of Exercises 11–16 shows the graph of a function y = ƒ(x).Copy the graph and draw in the line y = x. Then use symmetry withrespect to the line y = x to add the graph of ƒ -1 to your sketch. (It isnot necessary to find a formula for ƒ -1.) Identify the domain andrange of ƒ -1.
Calculus 11th Edition - Ron Larson
Chapter 4.4 - The Fundamental Theorem of Calculus
For the given problem, evaluate the definite integral. Please use a graphingutility to verify your result and post here. Please show work & explain steps, thank you.
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