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9. Evaluate the definite integral,

9. Evaluate the definite integral,

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter6: Exponential And Logarithmic Functions
Section6.7: Exponential And Logarithmic Models
Problem 27SE: Prove that bx=exln(b) for positive b1 .
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9. Evaluate the definite integral, x(1 – x)193dx 15
Transcribed Image Text:9. Evaluate the definite integral, x(1 – x)193dx 15
FUNDAMENTAL THEOREM OF CALCULUS I If fis continuous on [a,b] then the function BASIC ANTIDERIVATIVES Constant = ax +C, where a ER %3D Power: is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x). (rds = +C, where r + - 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then = In |x|+C Exponential: f(x)dx = F(b) – F(a). b +C, where b E (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)dx = sin(x) +C If F' is continuous on [a, b), then Jamer- sec-(endx = tan(x) + C = - cos(x) +C F(x)dx = F(b) – F(a). VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) + C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Joscondx = - cot(2) + C r)cot(x)dx = - csc(x) +C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS tan(x)dx = - In|cos(x)|+C_ If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jcot(o)d x = In |sin(<)| + c f(u)du. sec(x)dx = In|sec(x) + tan(x)| + C gla) x)dx = In[cse(x) – cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- xp = arcsin(x) + C, where x *±1 -ахis axis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C 1+x2 Cylindrical Shells Integrate y variable Integrate x variable Нурerbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on [a, b], then sinh(x) +C = cosh(x) + C Savg = f(x)dx. b- a
Transcribed Image Text:FUNDAMENTAL THEOREM OF CALCULUS I If fis continuous on [a,b] then the function BASIC ANTIDERIVATIVES Constant = ax +C, where a ER %3D Power: is continuous on [a, b], dfferentiable on (a, b) and gʻ'(x) = f(x). (rds = +C, where r + - 1 r +1 FUNDAMENTAL THEOREM OF CALCULUS II If fis continuous on [a,b] and F is any antiderivative of f, then = In |x|+C Exponential: f(x)dx = F(b) – F(a). b +C, where b E (0,00) In(b) Trigonometric: NET CHANGE THEOREM cos(x)dx = sin(x) +C If F' is continuous on [a, b), then Jamer- sec-(endx = tan(x) + C = - cos(x) +C F(x)dx = F(b) – F(a). VARIABLE SUBSTITUTION sec(x)tan(x)d x = sec(x) + C If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Joscondx = - cot(2) + C r)cot(x)dx = - csc(x) +C VARIABLE SUBSTITUTION FOR DEFINITE INTEGRALS tan(x)dx = - In|cos(x)|+C_ If u = g(x) is differentiable whose range contains [a, b] and fis continuous on [a, b), then Jcot(o)d x = In |sin(<)| + c f(u)du. sec(x)dx = In|sec(x) + tan(x)| + C gla) x)dx = In[cse(x) – cot(x)| +C VOLUMES OF SOLIDS OF REVOLUTION Inverse Trigonometric: Revolving about x Revolving about y- xp = arcsin(x) + C, where x *±1 -ахis axis Disks/Washers Integrate x variable Integrate y variable dx = arctan(x) + C 1+x2 Cylindrical Shells Integrate y variable Integrate x variable Нурerbolic AVERAGE VALUE OF A FUNCTION cosh(x)d x = sinh(x) +C Iffis continuous on [a, b], then sinh(x) +C = cosh(x) + C Savg = f(x)dx. b- a
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