Solve ∫ (from - 1 to 1) of (x^2)-(1)d

$$\int _ { - 1 } ^ { 1 } ( x ^ { 2 } ) - ( 1 ) d$$

Evaluate

$$\frac{2}{3}-2d$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-d\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -d\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-\int d\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-\int d\mathrm{d}x$$

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

$$\frac{x^{3}}{3}-dx$$

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.

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

Simplify.

$$\frac{2}{3}-2d$$

Differentiate w.r.t. d

$$-2$$