Solve ∫ 6q+6c+6x wrt x

$$\int 6 q + 6 c + 6 x d x$$

Evaluate

$$3x^{2}+6cx+6qx+С$$

Solution Steps

Integrate the sum term by term.

$$\int 6q\mathrm{d}x+\int 6c\mathrm{d}x+\int 6x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\int q\mathrm{d}x+6\int c\mathrm{d}x+6\int x\mathrm{d}x$$

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

$$6qx+6\int c\mathrm{d}x+6\int x\mathrm{d}x$$

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

$$6qx+6cx+6\int x\mathrm{d}x$$

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

$$6qx+6cx+3x^{2}$$

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.

$$6qx+6cx+3x^{2}+С$$

Differentiate w.r.t. x

$$6\left(x+c+q\right)$$