Solve ∫ (from -3 to 1) of 3x^2+8 wrt x

$$\int_{ -3 }^{ 1 } 3 { x }^{ 2 } +8 d x$$

Evaluate

$$60$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{2}+8\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{2}\mathrm{d}x+\int 8\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{2}\mathrm{d}x+\int 8\mathrm{d}x$$

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}.

$$x^{3}+\int 8\mathrm{d}x$$

Find the integral of 8 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{3}+8x$$

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.

$$1^{3}+8\times 1-\left(\left(-3\right)^{3}+8\left(-3\right)\right)$$

Simplify.

$$60$$