Solve ∫ (from 0 to 1) of 11sqrt[11]{x wrt x

$$\int _ { 0 } ^ { 1 } 11 \sqrt[ 11 ] { x } d x$$

Evaluate

$$\frac{121}{12}\approx 10.083333333$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 11\sqrt[11]{x}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$11\int \sqrt[11]{x}\mathrm{d}x$$

Rewrite \sqrt[11]{x} as x^{\frac{1}{11}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{11}}\mathrm{d}x with \frac{x^{\frac{12}{11}}}{\frac{12}{11}}. Simplify.

$$\frac{121x^{\frac{12}{11}}}{12}$$

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.

$$\frac{121}{12}\times 1^{\frac{12}{11}}-\frac{121}{12}\times 0^{\frac{12}{11}}$$

Simplify.

$$\frac{121}{12}$$