Solve ∫ 45x^2+2x+7 wrt x

$$\int{ 45 { x }^{ 2 } +2x+7 }d x$$

Evaluate

$$15x^{3}+x^{2}+7x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 45x^{2}\mathrm{d}x+\int 2x\mathrm{d}x+\int 7\mathrm{d}x$$

Factor out the constant in each of the terms.

$$45\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int 7\mathrm{d}x$$

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 45 times \frac{x^{3}}{3}.

$$15x^{3}+2\int x\mathrm{d}x+\int 7\mathrm{d}x$$

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

$$15x^{3}+x^{2}+\int 7\mathrm{d}x$$

Find the integral of 7 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$15x^{3}+x^{2}+7x$$

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.

$$15x^{3}+x^{2}+7x+С$$

Differentiate w.r.t. x

$$45x^{2}+2x+7$$