Solve ∫ (from 1 to 8) of (1/sqrt{x}-3x+5) wrt x=

$$\int _ { 1 } ^ { 8 } ( \frac { 1 } { \sqrt { x } } - 3 x + 5 ) d x =$$

Evaluate

$$4\sqrt{2}-\frac{123}{2}\approx -55.843145751$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{\sqrt{x}}-3x+5\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x+\int -3x\mathrm{d}x+\int 5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x-3\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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.

$$2\sqrt{x}-3\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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

$$2\sqrt{x}-\frac{3x^{2}}{2}+\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2\sqrt{x}-\frac{3x^{2}}{2}+5x$$

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.

$$2\times 8^{\frac{1}{2}}-\frac{3}{2}\times 8^{2}+5\times 8-\left(2\times 1^{\frac{1}{2}}-\frac{3}{2}\times 1^{2}+5\times 1\right)$$

Simplify.

$$4\sqrt{2}-\frac{123}{2}$$