Calculus

# L'Hopital's Rule - Convergence of Improper Integrals

What is the value of $\displaystyle{ \int_{\frac{1}{3}}^{\infty }\frac{1-5 \log 3x}{x^6} \mathrm{d} x}?$

What is the value of $\displaystyle{ \int_{0}^{3 } x^x (1+ \log x) \mathrm{d} x}?$

What is the value of $\displaystyle{ \int_{9}^{\infty }\left( \sin \frac{1}{4x}- \frac{1}{4x}\cos \frac{1}{4x} \right) \mathrm{d} x}?$

What is the value of $\displaystyle{ \int_{0}^{ \frac{\pi}{2} } \left( \left(\frac{\pi}{2} - x \right) \sec ^2 x -\tan x \right) \mathrm{d} x}?$

What is the value of $\displaystyle{ \int_{0}^{3 } \left( \frac{x-5(1+x)\log (1+x)}{(1+x)x^6} \right) \mathrm{d} x}?$

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