Solve ∫ wrt x/sqrt[3]{x^7}

$$\int \frac { d x } { \sqrt[ 3 ] { x ^ { 7 } } }$$

Evaluate

$$-\frac{3}{4x^{\frac{4}{3}}}+С$$

Solution Steps

Rewrite \frac{1}{x^{\frac{7}{3}}} as x^{-\frac{7}{3}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{7}{3}}\mathrm{d}x with \frac{x^{-\frac{4}{3}}}{-\frac{4}{3}}. Simplify and convert from exponential to radical form.

$$-\frac{3}{4x^{\frac{4}{3}}}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-\frac{3}{4x^{\frac{4}{3}}}+С$$

Differentiate w.r.t. x

$$\frac{1}{x^{\frac{7}{3}}}$$