Solve ∫ (from 5 to 7) of (4x^3-2sqrt[3]{x}+8/x^3) wrt x=

$$\int _ { 5 } ^ { 7 } ( 4 x ^ { 3 } - 2 \sqrt[ 3 ] { x } + \frac { 8 } { x ^ { 3 } } ) d x =$$

Evaluate

$$\frac{15\sqrt[3]{5}}{2}-\frac{21\sqrt[3]{7}}{2}+\frac{2175696}{1225}\approx 1768.817409528$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{3}-2\sqrt[3]{x}+\frac{8}{x^{3}}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{3}\mathrm{d}x+\int -2\sqrt[3]{x}\mathrm{d}x+\int \frac{8}{x^{3}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{3}\mathrm{d}x-2\int \sqrt[3]{x}\mathrm{d}x+8\int \frac{1}{x^{3}}\mathrm{d}x$$

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 4 times \frac{x^{4}}{4}.

$$x^{4}-2\int \sqrt[3]{x}\mathrm{d}x+8\int \frac{1}{x^{3}}\mathrm{d}x$$

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. Multiply -2 times \frac{3x^{\frac{4}{3}}}{4}.

$$x^{4}-\frac{3x^{\frac{4}{3}}}{2}+8\int \frac{1}{x^{3}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{3}}\mathrm{d}x with -\frac{1}{2x^{2}}. Multiply 8 times -\frac{1}{2x^{2}}.

$$x^{4}-\frac{3x^{\frac{4}{3}}}{2}-\frac{4}{x^{2}}$$

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.

$$7^{4}-\frac{3}{2}\times 7^{\frac{4}{3}}-4\times 7^{-2}-\left(5^{4}-\frac{3}{2}\times 5^{\frac{4}{3}}-4\times 5^{-2}\right)$$

Simplify.

$$\frac{2175696}{1225}-\frac{21\sqrt[3]{7}}{2}+\frac{15\sqrt[3]{5}}{2}$$