Answer
\[ = \frac{4}{3}\,\left( {{{10}^{\frac{3}{4}}}} \right)\]
Work Step by Step
\[\begin{gathered}
\int_0^{10} {\frac{{dx}}{{\sqrt[4]{{10 - x}}}}} \hfill \\
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integrate\,\,of\,\,indefinite\,\,integral \hfill \\
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set \hfill \\
10 - x = u\,\,\,\,then\,\,\,\,\, - dx = du \hfill \\
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therefore \hfill \\
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\int_{}^{} {\frac{{dx}}{{\sqrt[4]{{10 - x}}}}\,\, = \,\, - \,\,\int_{}^{} {\frac{1}{{{u^{\frac{1}{4}}}}}} \,du\,} \hfill \\
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{\text{integrate}}\,\,u\sin g\,\,the\,\,power\,\,rule \hfill \\
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= - \frac{4}{3}{u^{\frac{3}{4}}}\, \hfill \\
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= - \frac{4}{3}\,{\left( {10 - x} \right)^{\frac{3}{4}}} \hfill \\
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use\,\,\int_a^b {f\,\left( x \right)} \,dx\,\, = \,\,\mathop {\lim }\limits_{c \to {a^ + }} \int_c^b {f\,\left( x \right)} \,dx \hfill \\
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provided\,\,the\,\,\,limit\,\,exists \hfill \\
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\int_0^{10} {\frac{{dx}}{{\sqrt[4]{{10 - x}}}}} \,\,\, = \,\,\mathop {\lim }\limits_{a \to {{10}^ - }} \,\int_0^a {\frac{{dx}}{{\sqrt[4]{{10 - x}}}}} \hfill \\
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= \,\mathop {\lim }\limits_{a \to {{10}^ - }} \,\,\,\,\left[ { - \frac{4}{3}\,{{\left( {10 - x} \right)}^{\frac{3}{4}}}} \right]_0^a \hfill \\
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use\,\,the\,\,ftc \hfill \\
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= - \frac{4}{3}\mathop {\lim }\limits_{a \to {{10}^ - }} \,\,\,\left[ {\,{{\left( {10 - a} \right)}^{\frac{3}{4}}} - {{10}^{\frac{3}{4}}}} \right] \hfill \\
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simplify \hfill \\
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= \frac{4}{3}\,\left( {{{10}^{\frac{3}{4}}}} \right) \hfill \\
\end{gathered} \]
