Solve ∫ (from -1 to 1) of 3x^2+x+1 wrt x

$$\int_{ -1 }^{ 1 } 3 { x }^{ 2 } +x+1 d x$$

Evaluate

$$4$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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}.

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

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}.

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

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}+\frac{x^{2}}{2}+x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$1^{3}+\frac{1^{2}}{2}+1-\left(\left(-1\right)^{3}+\frac{\left(-1\right)^{2}}{2}-1\right)$$

Simplify.

$$4$$