Solve ∫ (from 1 to 4) of (-4/7sqrt{x}+3/4sqrt{x^3}-sqrt[4]{x5}+3) wrt x

$$\int_{ 1 }^{ 4 } (- \frac{ 4 }{ 7 } \sqrt{ x } + \frac{ 3 }{ 4 } \sqrt{ { x }^{ 3 } } - \sqrt[ 4 ]{ x5 } +3) d x$$

Evaluate

$$-3\sqrt[4]{x_{5}}+\frac{469}{30}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -\frac{4\sqrt{x}}{7}+\frac{3\sqrt{x^{3}}}{4}-\sqrt[4]{x_{5}}+3\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{4\sqrt{x}}{7}\mathrm{d}x+\int \frac{3x^{\frac{3}{2}}}{4}\mathrm{d}x+\int -\sqrt[4]{x_{5}}\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{4\int \sqrt{x}\mathrm{d}x}{7}+\frac{3\int x^{\frac{3}{2}}\mathrm{d}x}{4}-\int \sqrt[4]{x_{5}}\mathrm{d}x+\int 3\mathrm{d}x$$

Rewrite \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{3}{2}}}{\frac{3}{2}}. Simplify. Multiply -\frac{4}{7} times \frac{2x^{\frac{3}{2}}}{3}.

$$-\frac{8x^{\frac{3}{2}}}{21}+\frac{3\int x^{\frac{3}{2}}\mathrm{d}x}{4}-\int \sqrt[4]{x_{5}}\mathrm{d}x+\int 3\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{3}{2}}\mathrm{d}x with \frac{2x^{\frac{5}{2}}}{5}. Multiply \frac{3}{4} times \frac{2x^{\frac{5}{2}}}{5}.

$$-\frac{8x^{\frac{3}{2}}}{21}+\frac{3x^{\frac{5}{2}}}{10}-\int \sqrt[4]{x_{5}}\mathrm{d}x+\int 3\mathrm{d}x$$

Find the integral of \sqrt[4]{x_{5}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{8x^{\frac{3}{2}}}{21}+\frac{3x^{\frac{5}{2}}}{10}-\sqrt[4]{x_{5}}x+\int 3\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{8x^{\frac{3}{2}}}{21}+\frac{3x^{\frac{5}{2}}}{10}-\sqrt[4]{x_{5}}x+3x$$

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.

$$-\frac{8}{21}\times 4^{\frac{3}{2}}+\frac{3}{10}\times 4^{\frac{5}{2}}-x_{5}^{\frac{1}{4}}\times 4+3\times 4-\left(-\frac{8}{21}\times 1^{\frac{3}{2}}+\frac{3}{10}\times 1^{\frac{5}{2}}-x_{5}^{\frac{1}{4}}+3\times 1\right)$$

Simplify.

$$\frac{469}{30}-3\sqrt[4]{x_{5}}$$

Differentiate w.r.t. x_5

$$-\frac{3}{4x_{5}^{\frac{3}{4}}}$$