# Evaluate the following integral. \int_{1}^{32}(x^{-\frac{6}{5}})dx

Evaluate the following integral.
${\int }_{1}^{32}\left({x}^{-\frac{6}{5}}\right)dx$
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${\int }_{1}^{32}\left({x}^{-\frac{6}{5}}\right)dx={\left[\frac{{x}^{-\frac{6}{5}+1}}{-\frac{6}{5}+1}\right]}_{1}^{32}={\left[\frac{{x}^{\frac{-1}{5}}}{\frac{-1}{5}}\right]}_{1}^{32}$
$=\left(-5\right)\left[{\left(32\right)}^{\frac{-1}{5}}-{\left(1\right)}^{\frac{-1}{5}}\right]$
$=-5\left[{\left({2}^{5}\right)}^{\frac{-1}{5}}-1\right]=-5\left({2}^{-1}-1\right)$
Hence, ${\int }_{1}^{32}\left({x}^{\frac{-6}{5}}\right)dx=-5\left(\frac{1}{2}-1\right)=-5\left(\frac{-1}{2}\right)=\frac{5}{2}$