Solve ∫ (from 0 to 6) of (1/3)^x

$$\int _ { 0 } ^ { 6 } ( \frac { 1 } { 3 } ) ^ { x }$$

Evaluate

$$\frac{728}{729\ln(3)}\approx 0.908990613$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(\frac{1}{3}\right)^{x}\mathrm{d}x$$

Use \int x^{y}\mathrm{d}y=\frac{x^{y}}{\ln(x)} from the table of common integrals to obtain the result.

$$\frac{\left(\frac{1}{3}\right)^{x}}{\ln(\frac{1}{3})}$$

Simplify.

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

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.

$$-\left(\frac{1}{3}\right)^{6}\ln(3)^{-1}+\left(\frac{1}{3}\right)^{0}\ln(3)^{-1}$$

Simplify.

$$\frac{728}{729\ln(3)}$$