Solve ∫ (from 0 to 6) of 3^3x/9^-3x

$$\int _ { 0 } ^ { 6 } \frac { 3 ^ { 3 } x } { 9 ^ { - 3 } x }$$

Evaluate

$$118098$$

Solution Steps

Cancel out x in both numerator and denominator.

$$\int _{0}^{6}\frac{3^{3}}{9^{-3}}\mathrm{d}x$$

Calculate 3 to the power of 3 and get 27.

$$\int _{0}^{6}\frac{27}{9^{-3}}\mathrm{d}x$$

Calculate 9 to the power of -3 and get \frac{1}{729}.

$$\int _{0}^{6}\frac{27}{\frac{1}{729}}\mathrm{d}x$$

Divide 27 by \frac{1}{729} by multiplying 27 by the reciprocal of \frac{1}{729}.

$$\int _{0}^{6}27\times 729\mathrm{d}x$$

Multiply 27 and 729 to get 19683.

$$\int _{0}^{6}19683\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 19683\mathrm{d}x$$

Find the integral of 19683 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$19683x$$

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.

$$19683\times 6-19683\times 0$$

Simplify.

$$118098$$