Solve ∫ sqrt[5]{x^3}+sqrt{x^7}+2x^6) wrt x

$$\int \sqrt[ 5 ] { x ^ { 3 } } + \sqrt { x ^ { 7 } } + 2 x ^ { 6 } ) d x$$

Evaluate

$$\frac{2x^{7}}{7}+\frac{2x^{\frac{9}{2}}}{9}+\frac{5x^{\frac{8}{5}}}{8}+С$$

Solution Steps

Integrate the sum term by term.

$$\int x^{\frac{3}{5}}\mathrm{d}x+\int x^{\frac{7}{2}}\mathrm{d}x+\int 2x^{6}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{\frac{3}{5}}\mathrm{d}x+\int x^{\frac{7}{2}}\mathrm{d}x+2\int x^{6}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{3}{5}}\mathrm{d}x with \frac{5x^{\frac{8}{5}}}{8}.

$$\frac{5x^{\frac{8}{5}}}{8}+\int x^{\frac{7}{2}}\mathrm{d}x+2\int x^{6}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{7}{2}}\mathrm{d}x with \frac{2x^{\frac{9}{2}}}{9}.

$$\frac{5x^{\frac{8}{5}}}{8}+\frac{2x^{\frac{9}{2}}}{9}+2\int x^{6}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply 2 times \frac{x^{7}}{7}.

$$\frac{5x^{\frac{8}{5}}}{8}+\frac{2x^{\frac{9}{2}}}{9}+\frac{2x^{7}}{7}$$

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.

$$\frac{5x^{\frac{8}{5}}}{8}+\frac{2x^{\frac{9}{2}}}{9}+\frac{2x^{7}}{7}+С$$

Differentiate w.r.t. x

$$2x^{6}+x^{\frac{7}{2}}+x^{\frac{3}{5}}$$