Solve ∫ (from 1 to 2) of int_0^2dv/ln5

$$\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { 2 } \frac { d v } { \ln 5 }$$

Evaluate

$$\frac{2}{\ln(5)}\approx 1.242669869$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{2}\frac{1}{\log_{e}\left(5\right)}\mathrm{d}v\mathrm{d}x$$

Find the integral of \int _{0}^{2}\frac{1}{\log_{e}\left(5\right)}\mathrm{d}v using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\int _{0}^{2}\frac{1}{\log_{e}\left(5\right)}\mathrm{d}vx$$

Simplify.

$$\frac{2x}{\ln(5)}$$

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.

$$2\ln(5)^{-1}\times 2-2\ln(5)^{-1}$$

Simplify.

$$\frac{2}{\ln(5)}$$