Evaluate
\frac{6\left(6^{\frac{7}{6}}-1\right)}{7}\approx 6.075460224
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\int \sqrt[6]{x}\mathrm{d}x
Evaluate the indefinite integral first.
\frac{6x^{\frac{7}{6}}}{7}
Rewrite \sqrt[6]{x} as x^{\frac{1}{6}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{6}}\mathrm{d}x with \frac{x^{\frac{7}{6}}}{\frac{7}{6}}. Simplify.
\frac{6}{7}\times 6^{\frac{7}{6}}-\frac{6}{7}\times 1^{\frac{7}{6}}
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
\frac{36\sqrt[6]{6}-6}{7}
Simplify.
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