Solve a ^ - 1 ( 2 ) = 1,000 e ^ - ∫ (from 0 to 2) of ( 0.03+0.02r)dr

$$a ^ { - 1 } ( 2 ) = 1,000 e ^ { - \int _ { 0 } ^ { 2 } ( 0.03 + 0.02 r ) d r }$$

Solve for a

$$a=\frac{\sqrt[10]{e}}{500}\approx 0.002210342$$

Steps for Solving Linear Equation

Reorder the terms.

$$2\times \frac{1}{a}=1000e^{-\int _{0}^{2}0.02r+0.03\mathrm{d}r}$$

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$2\times 1=1000e^{-\int _{0}^{2}0.02r+0.03\mathrm{d}r}a$$

Multiply 2 and 1 to get 2.

$$2=1000e^{-\int _{0}^{2}0.02r+0.03\mathrm{d}r}a$$

Swap sides so that all variable terms are on the left hand side.

$$1000e^{-\int _{0}^{2}0.02r+0.03\mathrm{d}r}a=2$$

The equation is in standard form.

$$\frac{1000}{\sqrt[10]{e}}a=2$$

Divide both sides by 1000e^{-\frac{1}{10}}.

$$\frac{\frac{1000}{\sqrt[10]{e}}a\sqrt[10]{e}}{1000}=\frac{2\sqrt[10]{e}}{1000}$$

Dividing by 1000e^{-\frac{1}{10}} undoes the multiplication by 1000e^{-\frac{1}{10}}.

$$a=\frac{2\sqrt[10]{e}}{1000}$$

Divide 2 by 1000e^{-\frac{1}{10}}.

$$a=\frac{\sqrt[10]{e}}{500}$$

Variable a cannot be equal to 0.

$$a=\frac{\sqrt[10]{e}}{500}\text{, }a\neq 0$$