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First make a substitution and then use integration by parts to evaluate the integral.?

0 votes
First make a substitution and then use integration by parts to evaluate the integral.
∫ t^15 e^-t^8 dt
asked Apr 24, 2013 in CALCULUS by Jose Rodriguez Rookie

2 Answers

0 votes

integral t to the power of 15 open parentheses e close parentheses to the power of minus t to the power of 8 end exponent d t equal integral t to the power of 8 cross times t to the power of 7 open parentheses e close parentheses to the power of minus t to the power of 8 end exponent d t   

space space space space space space space space space space space space space space space space space space space space equal integral x cross times t to the power of 7 d t open parentheses e close parentheses to the power of minus x end exponent      (L e t space t to the power of 8 equal x space t h e n space 8 t to the power of 7 d t equal d x)

space space space space space space space space space space space space space space space space space space space space equal integral x cross times 1 over 8 d x open parentheses e close parentheses to the power of minus x end exponent   (Substitute t to the power of 7 d t equal 1 over 8 d x)

space space space space space space space space space space space space space space space space space space space space equal 1 over 8 integral x cross times open parentheses e close parentheses to the power of minus x end exponent d x    

space space space space space space space space space space space space space space space space space space space space equal 1 over 8 open square brackets x integral open parentheses e close parentheses to the power of minus x end exponent d x minus integral open parentheses 1 x to the power of 0 integral open parentheses e close parentheses to the power of minus x end exponent d x close parentheses d x close square brackets

                                                     (integral f open parentheses x close parentheses g open parentheses x close parentheses d x equal f open parentheses x close parentheses integral g open parentheses x close parentheses d x minus integral open parentheses d open parentheses f open parentheses x close parentheses close parentheses integral g open parentheses x close parentheses d x close parentheses d x right parenthesis

space space space space space space space space space space space space space space space space space space space space equal 1 over 8 open square brackets x open parentheses minus e to the power of minus x end exponent close parentheses minus integral minus e to the power of minus x end exponent d x close square brackets

space space space space space space space space space space space space space space space space space space space space equal 1 over 8 open square brackets x open parentheses minus e to the power of minus x end exponent close parentheses minus e to the power of minus x end exponent close square brackets

space space space space space space space space space space space space space space space space space space space space equal fraction numerator minus e to the power of minus x end exponent over denominator 8 end fraction open square brackets x plus 1 close square brackets

space space space space space space space space space space space space space space space space space space space space equal fraction numerator minus e to the power of minus t to the power of 8 end exponent over denominator 8 end fraction open square brackets t to the power of 8 plus 1 close square brackets      (Substitute x equal t to the power of 8)

 

 


 

answered Apr 24, 2013 by diane Scholar
0 votes

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Substitution method

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Now the integral becomes image

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By parts of integration formulaimage

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Again substitute image

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answered Jul 9, 2014 by david Expert

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