Solve ∫ (from 10 to 20) of (x^2-1)e^-02x wrt x

$$\int _ { 10 } ^ { 20 } ( x ^ { 2 } - 1 ) e ^ { - 02 x } d x$$

Evaluate

$$\frac{6970}{3}\approx 2323.333333333$$

Solution Steps

Multiply 0 and 2 to get 0.

$$\int _{10}^{20}\left(x^{2}-1\right)e^{0x}\mathrm{d}x$$

Anything times zero gives zero.

$$\int _{10}^{20}\left(x^{2}-1\right)e^{0}\mathrm{d}x$$

Calculate e to the power of 0 and get 1.

$$\int _{10}^{20}\left(x^{2}-1\right)\times 1\mathrm{d}x$$

Use the distributive property to multiply x^{2}-1 by 1.

$$\int _{10}^{20}x^{2}-1\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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

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

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.

$$\frac{20^{3}}{3}-20-\left(\frac{10^{3}}{3}-10\right)$$

Simplify.

$$\frac{6970}{3}$$