Evaluate the indefinite integral.
(Use symbolic notation and fractions where needed. Use $\mathrm{C}$ for the arbitrary constant. Absorb into $\mathrm{C}$ as much as possible.)
\[
\int 52 x^{-1 / 5} \tan \left(x^{4 / 5}\right) d x=
\]

Answer

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Answer

The indefinite integral of $52 x^{-1 / 5} \tan \left(x^{4 / 5}\right) d x$ is $\boxed{-65.0 \log(\cos(x^{0.8})) + C}$

Steps

Step 1 ：Define the function $f = 52 x^{-1 / 5} \tan \left(x^{4 / 5}\right)$

Step 2 ：Calculate the indefinite integral of the function $f$

Step 3 ：The output from the calculation is the integral of the function, which is $-65.0 \log(\cos(x^{0.8}))$

Step 4 ：However, we need to add the constant of integration, $C$, to the result

Step 5 ：The indefinite integral of $52 x^{-1 / 5} \tan \left(x^{4 / 5}\right) d x$ is $\boxed{-65.0 \log(\cos(x^{0.8})) + C}$