- #1
unique_pavadrin
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Homework Statement
Differentiate the following expression leaving in the simplest form:
[tex]\left( {7x^3 + 4} \right)^{\frac{1}{x}} [/tex]
Integrate the following leaving in the simplest form:
[tex]x^x \left( {1 + \ln x} \right)[/tex]
2. The attempt at a solution
Here is my worked solution for the differentiation problem:
[tex]\[
\begin{array}{l}
\frac{d}{{dx}}\left[ {\left( {7x^3 + 4} \right)^{\frac{1}{x}} } \right] \\
y = \left( {7x^3 + 4} \right)^{\frac{1}{x}} \\
\ln y = \left( {\frac{1}{x}} \right)\ln \left( {7x^3 + 4} \right) \\
= \frac{{\ln \left( {7x^3 + 4} \right)}}{x} \\
\frac{d}{{dx}}\left[ {\ln \left( {7x^3 + 4} \right)} \right] = \frac{1}{{7x^3 + 4}}.\frac{d}{{dx}}\left[ {7x^3 + 4} \right] \\
= \frac{{21x^2 }}{{7x^3 + 4}} \\
\frac{d}{{dx}}\left[ {\frac{{\ln \left( {7x^3 + 4} \right)}}{x}} \right] = \frac{{x\left( {\frac{{21x^2 }}{{7x^3 + 4}}} \right) + 1\left( {\ln \left( {7x^3 + 4} \right)} \right)}}{{x^2 }} \\
= \frac{{21x^3 + \left( {7x^3 + 4} \right)\ln \left( {7x^3 + 4} \right)}}{{x^2 }} \\
\\
\frac{1}{y} = \\
y' = y.\frac{{21x^3 + \left( {7x^3 + 4} \right)\ln \left( {7x^3 + 4} \right)}}{{x^2 \left( {7x^3 + 4} \right)}} \\
= \frac{{\left( {7x^3 + 4} \right)^{\frac{1}{x}} \left( {21x^3 + \left( {7x^3 + 4} \right)\ln \left( {7x^3 + 4} \right)} \right)}}{{x^2 \left( {7x^3 + 4} \right)}} \\
= \frac{{\left( {7x^3 + 4} \right)^{\frac{1}{x} - 1} \left( {21x^3 + \left( {7x^3 + 4} \right)\ln \left( {7x^3 + 4} \right)} \right)}}{{x^2 }} \\
\end{array}
\]
[/tex]
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For the integration problem I am not quite certain on how to integrate the expression given. I know from previous experience that the expression x^x when differentiated will give x^x(1+ln(x)).
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many thanks for all suggestions and help
unique_pavadrin