Solve ∫ (from 1 to 3) of (2+sqrt[3]{x}+sqrt[4]{x^3}+3x^3) wrt x

$$\int _ { 1 } ^ { 3 } ( 2 + \sqrt[ 3 ] { x } + \sqrt[ 4 ] { x ^ { 3 } } + 3 x ^ { 3 } ) d x$$

Evaluate

$$\frac{9\sqrt[3]{3}}{4}+\frac{12\times 3^{\frac{3}{4}}}{7}+\frac{1755}{28}\approx 69.831359345$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2+\sqrt[3]{x}+\sqrt[4]{x^{3}}+3x^{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2\mathrm{d}x+\int \sqrt[3]{x}\mathrm{d}x+\int x^{\frac{3}{4}}\mathrm{d}x+\int 3x^{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 2\mathrm{d}x+\int \sqrt[3]{x}\mathrm{d}x+\int x^{\frac{3}{4}}\mathrm{d}x+3\int x^{3}\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x+\int \sqrt[3]{x}\mathrm{d}x+\int x^{\frac{3}{4}}\mathrm{d}x+3\int x^{3}\mathrm{d}x$$

Rewrite \sqrt[3]{x} as x^{\frac{1}{3}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{3}}\mathrm{d}x with \frac{x^{\frac{4}{3}}}{\frac{4}{3}}. Simplify.

$$2x+\frac{3x^{\frac{4}{3}}}{4}+\int x^{\frac{3}{4}}\mathrm{d}x+3\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{3}{4}}\mathrm{d}x with \frac{4x^{\frac{7}{4}}}{7}.

$$2x+\frac{3x^{\frac{4}{3}}}{4}+\frac{4x^{\frac{7}{4}}}{7}+3\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 3 times \frac{x^{4}}{4}.

$$2x+\frac{3x^{\frac{4}{3}}}{4}+\frac{4x^{\frac{7}{4}}}{7}+\frac{3x^{4}}{4}$$

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.

$$2\times 3+\frac{3}{4}\times 3^{\frac{4}{3}}+\frac{4}{7}\times 3^{\frac{7}{4}}+\frac{3}{4}\times 3^{4}-\left(2\times 1+\frac{3}{4}\times 1^{\frac{4}{3}}+\frac{4}{7}\times 1^{\frac{7}{4}}+\frac{3}{4}\times 1^{4}\right)$$

Simplify.

$$\frac{1755}{28}+\frac{9\sqrt[3]{3}}{4}+\frac{12\times 3^{\frac{3}{4}}}{7}$$