Solve ∫ (from - 1 to 2) of (5-x^2-x^2+2x-1) wrt x

$$\int _ { - 1 } ^ { 2 } ( 5 - x ^ { 2 } - x ^ { 2 } + 2 x - 1 ) d x$$

Evaluate

$$9$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5-x^{2}-x^{2}+2x-1\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5\mathrm{d}x+\int -x^{2}\mathrm{d}x+\int -x^{2}\mathrm{d}x+\int 2x\mathrm{d}x+\int -1\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 5\mathrm{d}x-\int x^{2}\mathrm{d}x-\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -1\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$5x-\int x^{2}\mathrm{d}x-\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -1\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$5x-\frac{x^{3}}{3}-\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -1\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$5x-\frac{x^{3}}{3}-\frac{x^{3}}{3}+2\int x\mathrm{d}x+\int -1\mathrm{d}x$$

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

$$5x-\frac{x^{3}}{3}-\frac{x^{3}}{3}+x^{2}+\int -1\mathrm{d}x$$

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

$$5x-\frac{x^{3}}{3}-\frac{x^{3}}{3}+x^{2}-x$$

Simplify.

$$4x-\frac{2x^{3}}{3}+x^{2}$$

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.

$$4\times 2-\frac{2}{3}\times 2^{3}+2^{2}-\left(4\left(-1\right)-\frac{2}{3}\left(-1\right)^{3}+\left(-1\right)^{2}\right)$$

Simplify.

$$9$$