Solve ∫ (from 9 to 25) of (4-1/sqrt{x}) wrt x=

$$\int _ { 9 } ^ { 25 } ( 4 - \frac { 1 } { \sqrt { x } } ) d x =$$

Evaluate

$$60$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4-\frac{1}{\sqrt{x}}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4\mathrm{d}x+\int -\frac{1}{\sqrt{x}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 4\mathrm{d}x-\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$4x-\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Rewrite \frac{1}{\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{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form. Multiply -1 times 2\sqrt{x}.

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

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.

$$4\times 25-2\times 25^{\frac{1}{2}}-\left(4\times 9-2\times 9^{\frac{1}{2}}\right)$$

Simplify.

$$60$$