Solve ∫ 1/xln(1.5) wrt x

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

Evaluate

$$\log_{1.5}\left(|x|\right)+С$$

Solution Steps

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.

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

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

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

Differentiate w.r.t. x

$$\frac{1}{\ln(1.5)x}$$