Solve ∫ (from 0 to 25) of (-12sqrt{x}) wrt x=

$$\int _ { 0 } ^ { 25 } ( - 12 \sqrt { x } ) d x =$$

Evaluate

$$-1000$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -12\sqrt{x}\mathrm{d}x$$

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

$$-12\int \sqrt{x}\mathrm{d}x$$

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 -12 times \frac{2x^{\frac{3}{2}}}{3}.

$$-8x^{\frac{3}{2}}$$

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.

$$-8\times 25^{\frac{3}{2}}+8\times 0^{\frac{3}{2}}$$

Simplify.

$$-1000$$