Find Difficult Indefinite Integral ∫ ( log ⁡ x + 1 ) x x d...

Audrey Arnold

Audrey Arnold

Answered

2022-11-11

Find Difficult Indefinite Integral ( log x + 1 ) x x d x

Answer & Explanation

Lillianna Salazar

Lillianna Salazar

Expert

2022-11-12Added 22 answers

The form of the integrand suggests writing
( 1 + log x ) x x = ( 1 + log x ) e x log x ,
then observing that by the product rule,
d d x [ x log x ] = x 1 x + 1 log x = 1 + log x .
Consequently, the integrand is of the form f ( x ) e f ( x ) , and its antiderivative is simply
e f ( x ) = e x log x = x x .
Adison Rogers

Adison Rogers

Expert

2022-11-13Added 2 answers

Use the substitution y = x x , then do logarithmic differentitation. to get ( 1 + ln x ) x x = d y d x

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