Integral with Logarithmic Functions

132

$$\int{\ln{x}}\ \mathrm{d}x=x\ln{x}-x+C$$

Derivation

133

$$\int{\frac{1}{x\ln{x}}}\ \mathrm{d}x=\ln{\left|\ln{x}\right|}+C$$

Derivation

134

$$\int{x^{n}\ln{x}}\ \mathrm{d}x=\frac{1}{n+1}x^{n+1}\left(\ln{x}-\frac{1}{n+1} \right)+C$$

Derivation

135

$$\int{(\ln{x})^n}=x(\ln{x})^n-n\int{(\ln{x})^{n-1}}\ \mathrm{d}x$$

Derivation

136

$$\int{x^m(\ln{x})^n}\ \mathrm{d}x=\frac{1}{m+1}x^{m+1}(\ln{x})^n-\frac{n}{m+1}\int{x^m(\ln{x})^{n-1}}\ \mathrm{d}x$$

Derivation