Solve ∫ (from 0 to 5) of (sqrt{a}-sqrt{x}) wrt x

$$\int _ { 0 } ^ { 5 } ( \sqrt { a } - \sqrt { x } ) d x$$

Evaluate

$$5\sqrt{a}-\frac{10\sqrt{5}}{3}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

$$\int \sqrt{a}\mathrm{d}x+\int -\sqrt{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

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

Find the integral of \sqrt{a} using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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 -1 times \frac{2x^{\frac{3}{2}}}{3}.

$$\sqrt{a}x-\frac{2x^{\frac{3}{2}}}{3}$$

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.

$$a^{\frac{1}{2}}\times 5-\frac{2}{3}\times 5^{\frac{3}{2}}-\left(a^{\frac{1}{2}}\times 0-\frac{2}{3}\times 0^{\frac{3}{2}}\right)$$

Simplify.

$$5\sqrt{a}-\frac{10\sqrt{5}}{3}$$

Differentiate w.r.t. a

$$\frac{5}{2\sqrt{a}}$$