Exercícios Resolvidos
$a)f\left( x \right)=\left( 4x+6 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}\Rightarrow \int{\left( 4x+6 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}dx\Rightarrow }$
$\begin{matrix}
U \\
du \\
\end{matrix}\begin{matrix}
= \\
= \\
\end{matrix}\begin{matrix}
{{x}^{2}}-3x+1 \\
2x-3 \\
\end{matrix}$
$\Rightarrow \int{\frac{2}{2}.\left( 4x+6 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}dx\Rightarrow }$
$\Rightarrow \int{\frac{2}{1}.\left( 2x+3 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}dx\Rightarrow }$
$\Rightarrow 2\int{\left( 2x+3 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}dx\Rightarrow }$
$\Rightarrow \int{{{U}^{3}}du=\frac{{{U}^{3+1}}}{3+1}}+C=\frac{{{U}^{4}}}{4}+C\Rightarrow $
Logo:
$\Rightarrow 2\int{\left( 2x+3 \right){{\left( {{x}^{2}}-3x+1 \right)}^{3}}dx}=\frac{2{{\left( {{x}^{2}}-3x+1 \right)}^{4}}}{4}+C=$
$=\frac{{{\left( {{x}^{2}}-3x+1 \right)}^{4}}}{2}+C$
$b)f\left( x \right)=3{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}\Rightarrow \int{3{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}dx}\Rightarrow $
$\begin{matrix}
U \\
du \\
\end{matrix}\begin{matrix}
= \\
= \\
\end{matrix}\begin{matrix}
2-{{x}^{5}} \\
-5{{x}^{4}} \\
\end{matrix}$
$\Rightarrow \int{\frac{-5}{-5}.\frac{3}{3}3{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}dx}=\int{\frac{3}{-5}.-5{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}dx}\Rightarrow $
$\Rightarrow -\frac{3}{5}.\int{-5{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}dx}\Rightarrow $
$\Rightarrow \int{{{U}^{3}}du=\frac{{{U}^{3+1}}}{3+1}}+C=\frac{{{U}^{4}}}{4}+C\Rightarrow $
Logo:
$\Rightarrow -\frac{3}{5}.\int{-5{{x}^{4}}{{\left( 2-{{x}^{5}} \right)}^{3}}dx}=-\frac{3}{5}.\frac{{{\left( 2-{{x}^{5}} \right)}^{4}}}{4}+C=$
$=\frac{-3{{\left( 2-{{x}^{5}} \right)}^{4}}}{20}+C$
$c)f\left( x \right)=\left( \frac{3{{x}^{3}}}{2{{x}^{4}}-5} \right)\Rightarrow \int{\left( \frac{3{{x}^{3}}}{2{{x}^{4}}-5} \right)dx}\Rightarrow $
$\begin{matrix}
U \\
du \\
\end{matrix}\begin{matrix}
= \\
= \\
\end{matrix}\begin{matrix}
2{{x}^{4}}-5 \\
8{{x}^{3}} \\
\end{matrix}$
$\Rightarrow \int{\frac{8}{8}.\left( \frac{3{{x}^{3}}}{2{{x}^{4}}-5} \right)dx}\Rightarrow \int{\frac{3}{8}.\left( \frac{8{{x}^{3}}}{2{{x}^{4}}-5} \right)dx}\Rightarrow $
$\Rightarrow \frac{3}{8}\int{\left( \frac{8{{x}^{3}}}{2{{x}^{4}}-5} \right)dx}\Rightarrow $
$\Rightarrow \frac{3}{8}\int{\frac{du}{U}=\frac{3}{8}.\ln \left( u \right)}+C\Rightarrow $
Logo:
$\Rightarrow \frac{3}{8}\int{\left( \frac{8{{x}^{3}}}{2{{x}^{4}}-5} \right)dx}=\frac{3}{8}.\ln \left( 2{{x}^{4}}-5 \right)+C$
$d)f\left( x \right)=10x~sen~\left( {{x}^{2}}-7 \right)\Rightarrow \int{10x~sen~\left( {{x}^{2}}-7 \right)dx\Rightarrow }$
$\begin{matrix}
U \\
du \\
\end{matrix}\begin{matrix}
= \\
= \\
\end{matrix}\begin{matrix}
{{x}^{2}}-7 \\
2x \\
\end{matrix}$
$\Rightarrow \int{\frac{5}{5}.10x~sen~\left( {{x}^{2}}-7 \right)dx\Rightarrow }\int{\frac{5}{1}.2x~sen~\left( {{x}^{2}}-7 \right)dx\Rightarrow }$
$\Rightarrow 5\int{2x~sen~\left( {{x}^{2}}-7 \right)dx\Rightarrow }$
$\Rightarrow 5\int{sen\left( U \right)}~du=-5\cos \left( U \right)+C$
Logo:
$\Rightarrow 5\int{2x~sen~\left( {{x}^{2}}-7 \right)dx}=-5\cos \left( {{x}^{2}}-7 \right)+C$
$e)f\left( x \right)=\frac{8}{3}~{{e}^{4x}}\Rightarrow \int{\frac{8}{3}~{{e}^{4x}}}~dx$
$\begin{matrix}
U \\
du \\
\end{matrix}\begin{matrix}
= \\
= \\
\end{matrix}\begin{matrix}
4x \\
4 \\
\end{matrix}$
$\Rightarrow \int{\frac{2}{2}.\frac{3}{3}.\frac{8}{3}~{{e}^{4x}}}~dx\Rightarrow \int{\frac{2}{3}.4~{{e}^{4x}}}~dx\Rightarrow $
$\Rightarrow \frac{2}{3}\int{4~{{e}^{4x}}}~dx\Rightarrow $
$\Rightarrow \frac{2}{3}{{\int{e}}^{U}}du=\frac{2}{3}{{e}^{U}}+C$
Logo:
$\Rightarrow \frac{2}{3}\int{4~{{e}^{4x}}}~dx=\frac{2}{3}{{e}^{4x}}+C$
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