Solve ∫ (10x^5-x^3-14x^2-1/2x+3) wrt x

$$\int{ (10 { x }^{ 5 } - { x }^{ 3 } -14 { x }^{ 2 } - \frac{ 1 }{ 2 } x+3) }d x$$

Evaluate

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 10x^{5}\mathrm{d}x+\int -x^{3}\mathrm{d}x+\int -14x^{2}\mathrm{d}x+\int -\frac{x}{2}\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$10\int x^{5}\mathrm{d}x-\int x^{3}\mathrm{d}x-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 10 times \frac{x^{6}}{6}.

$$\frac{5x^{6}}{3}-\int x^{3}\mathrm{d}x-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-14\int x^{2}\mathrm{d}x-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -14 times \frac{x^{3}}{3}.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{\int x\mathrm{d}x}{2}+\int 3\mathrm{d}x$$

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

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+\int 3\mathrm{d}x$$

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

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x$$

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.

$$\frac{5x^{6}}{3}-\frac{x^{4}}{4}-\frac{14x^{3}}{3}-\frac{x^{2}}{4}+3x+С$$

Differentiate w.r.t. x

$$10x^{5}-x^{3}-14x^{2}-\frac{x}{2}+3$$