Solve ∫ (from 1 to 4) of 5sqrt[4]{x wrt x

$$\int _ { 1 } ^ { 4 } 5 \sqrt[ 4 ] { x } d x$$

Evaluate

$$16\sqrt{2}-4\approx 18.627416998$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5\sqrt[4]{x}\mathrm{d}x$$

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

$$5\int \sqrt[4]{x}\mathrm{d}x$$

Rewrite \sqrt[4]{x} as x^{\frac{1}{4}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{4}}\mathrm{d}x with \frac{x^{\frac{5}{4}}}{\frac{5}{4}}. Simplify. Multiply 5 times \frac{4x^{\frac{5}{4}}}{5}.

$$4x^{\frac{5}{4}}$$

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.

$$4\times 4^{\frac{5}{4}}-4\times 1^{\frac{5}{4}}$$

Simplify.

$$16\sqrt{2}-4$$