Solve ∫ 12x^2+16*2x+62*3 wrt x

$$\int 12 x ^ { 2 } + 16 \cdot 2 x + 62 \cdot 3 d x$$

Evaluate

$$4x^{3}+16x^{2}+186x+С$$

Solution Steps

Do the multiplications.

$$\int 12x^{2}+32x+186\mathrm{d}x$$

Integrate the sum term by term.

$$\int 12x^{2}\mathrm{d}x+\int 32x\mathrm{d}x+\int 186\mathrm{d}x$$

Factor out the constant in each of the terms.

$$12\int x^{2}\mathrm{d}x+32\int x\mathrm{d}x+\int 186\mathrm{d}x$$

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

$$4x^{3}+32\int x\mathrm{d}x+\int 186\mathrm{d}x$$

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

$$4x^{3}+16x^{2}+\int 186\mathrm{d}x$$

Find the integral of 186 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$4x^{3}+16x^{2}+186x$$

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.

$$4x^{3}+16x^{2}+186x+С$$

Differentiate w.r.t. x

$$12x^{2}+32x+186$$