Solve ∫ (from 4 to 16) of (3sqrt{x}) wrt x=

$$\int _ { 4 } ^ { 16 } ( 3 \sqrt { x } ) d x =$$

Evaluate

$$112$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3\sqrt{x}\mathrm{d}x$$

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

$$3\int \sqrt{x}\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply 3 times \frac{2x^{\frac{3}{2}}}{3}.

$$2x^{\frac{3}{2}}$$

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.

$$2\times 16^{\frac{3}{2}}-2\times 4^{\frac{3}{2}}$$

Simplify.

$$112$$