Solve ∫ (from 4 to 6) of 1/xln(1.5) wrt x

$$\int _ { 4 } ^ { 6 } \frac { 1 } { x \ln ( 1.5 ) } d x$$

Evaluate

$$1$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{x\log_{e}\left(1.5\right)}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{1}{\ln(1.5)\times \frac{1}{\ln(e)}}\int \frac{1}{x}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$\frac{1}{\ln(1.5)\times \frac{1}{\ln(e)}}\ln(|x|)$$

Simplify.

$$\frac{\ln(|x|)}{\ln(\frac{3}{2})}$$

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.

$$\left(\ln(3)-\ln(2)\right)^{-1}\ln(|6|)-\left(\ln(3)-\ln(2)\right)^{-1}\ln(|4|)$$

Simplify.

$$1$$