Solve ∫ (3/x^4+4/sqrt{x}-sqrt[4]{x^3}) wrt x

$$\int ( \frac { 3 } { x ^ { 4 } } + \frac { 4 } { \sqrt { x } } - \sqrt[ 4 ] { x ^ { 3 } } ) d x$$

Evaluate

$$-\frac{4x^{\frac{7}{4}}}{7}+8\sqrt{x}-\frac{1}{x^{3}}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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. Multiply 4 times 2\sqrt{x}.

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

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}. Multiply -1 times \frac{4x^{\frac{7}{4}}}{7}.

$$-\frac{1}{x^{3}}+8\sqrt{x}-\frac{4x^{\frac{7}{4}}}{7}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-\frac{1}{x^{3}}+8\sqrt{x}-\frac{4x^{\frac{7}{4}}}{7}+С$$

Differentiate w.r.t. x

$$-x^{\frac{3}{4}}+\frac{4}{\sqrt{x}}+\frac{3}{x^{4}}$$