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

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 .

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High School
Calculus and Analysis
Calculus 1
Logarithmic Functions

Audrey Arnold

Answered question

2022-11-11

Find Difficult Indefinite Integral

\int (\log x + 1) x^{x} d x

Answer & Explanation

Lillianna Salazar

Beginner2022-11-12Added 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,

\frac{d}{d x} [x \log x] = x \cdot \frac{1}{x} + 1 \cdot \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

Beginner2022-11-13Added 2 answers

Use the substitution

y = x^{x}

, then do logarithmic differentitation. to get

(1 + \ln x) x^{x} = \frac{d y}{d x}

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