Solve ∫ (-7sqrt{x}+5sqrt[4]{x}) wrt x

$$\int ( - 7 \sqrt { x } + 5 \sqrt[ 4 ] { x } ) d x$$

Evaluate

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 -7 times \frac{2x^{\frac{3}{2}}}{3}.

$$-\frac{14x^{\frac{3}{2}}}{3}+5\int \sqrt[4]{x}\mathrm{d}x$$

Rewrite \sqrt[4]{x} as x^{\frac{1}{4}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{4}}\mathrm{d}x with \frac{x^{\frac{5}{4}}}{\frac{5}{4}}. Simplify. Multiply 5 times \frac{4x^{\frac{5}{4}}}{5}.

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}$$

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.

$$-\frac{14x^{\frac{3}{2}}}{3}+4x^{\frac{5}{4}}+С$$

Differentiate w.r.t. x

$$-7\sqrt{x}+5\sqrt[4]{x}$$