Solve ∫ (3^3+3x^2+1) wrt x

$$\int ( 3 ^ { 3 } + 3 x ^ { 2 } + 1 ) d x$$

Evaluate

$$x^{3}+28x+С$$

Solution Steps

Calculate 3 to the power of 3 and get 27.

$$\int 27+3x^{2}+1\mathrm{d}x$$

Add 27 and 1 to get 28.

$$\int 28+3x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 28\mathrm{d}x+\int 3x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 28\mathrm{d}x+3\int x^{2}\mathrm{d}x$$

Find the integral of 28 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$28x+3\int x^{2}\mathrm{d}x$$

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

$$28x+x^{3}$$

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.

$$28x+x^{3}+С$$

Differentiate w.r.t. x

$$3x^{2}+28$$