Solve ∫ (from 1 to 3) of (sqrt{x^3}+1) wrt x

$$\int _ { 1 } ^ { 3 } ( \sqrt { x ^ { 3 } } + 1 ) d x$$

Evaluate

$$\frac{18\sqrt{3}+8}{5}\approx 7.835382907$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{x^{3}}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{\frac{3}{2}}\mathrm{d}x+\int 1\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{3}{2}}\mathrm{d}x with \frac{2x^{\frac{5}{2}}}{5}.

$$\frac{2x^{\frac{5}{2}}}{5}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{2x^{\frac{5}{2}}}{5}+x$$

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.

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

Simplify.

$$\frac{18\sqrt{3}+8}{5}$$