Math Problem Statement

finsih this integration

Finish the integration $ \int \frac{1}{x \ln(\sqrt{x}) - 9} \, dx $, with a hint $ e^{\ln x} = x $.

Solution

from sympy import symbols, integrate, log, sqrt

Define the variables

x = symbols('x')

Expression to integrate

expr = 1 / (x * (log(sqrt(x))) - 9)

Perform the integration

integral_result = integrate(expr, x) integral_result2Integral(1/(xlog(x) - 18), x)2Integral(1/(xlog(x) - 18), x)successThe integral simplifies to $2 \int \frac{1}{x \ln(x) - 18} \, dx$ . It seems further simplification or specific techniques are required for a closed-form solution.

Would you like me to delve deeper into this or explore numerical solutions?

Questions:

Should we focus on simplifying the logarithmic term further?
Do you want numerical integration for a specific range?
Shall I analyze the hint $e^{\ln x} = x$ for substitution?
Would you like a step-by-step explanation for this?
Should we graph the denominator to analyze its behavior?

Tip: Substitution methods often work well with logarithmic integrals.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Integration
Logarithmic Functions
Substitution

Formulas

$ \ln(\sqrt{x}) = \frac{1}{2} \ln(x) $
$ e^{\ln x} = x $

Theorems

Logarithmic properties
Substitution in definite integrals

Suitable Grade Level

College-level Calculus

Related Recommendation

Evaluate the Integral ∫[1,9] 1/(√x(1 + √x)) dx using Substitution

Evaluate Integral of (ln x + 1) / ((ln x)^2 (1 + x ln x))

Definite Integral of 1/(sqrt(x) + 9*cbrt(x)) from 0 to 1

Solving the Integral $ \int \frac{dx}{x \ln x} $ Using Substitution

Solve the Integral of ln(9x)/x^3 using Integration by Parts