Solve ∫ 14x^2+20x+25 wrt x

$$\int 14 x ^ { 2 } + 20 x + 25 d x$$

Evaluate

$$\frac{14x^{3}}{3}+10x^{2}+25x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 14x^{2}\mathrm{d}x+\int 20x\mathrm{d}x+\int 25\mathrm{d}x$$

Factor out the constant in each of the terms.

$$14\int x^{2}\mathrm{d}x+20\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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}.

$$\frac{14x^{3}}{3}+20\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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 20 times \frac{x^{2}}{2}.

$$\frac{14x^{3}}{3}+10x^{2}+\int 25\mathrm{d}x$$

Find the integral of 25 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{14x^{3}}{3}+10x^{2}+25x$$

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.

$$\frac{14x^{3}}{3}+10x^{2}+25x+С$$

Differentiate w.r.t. x

$$14x^{2}+20x+25$$