6.2.1 Power rule; $\int x^n\thinspace dx\thinspace$ & $\thinspace\int (1/x)\thinspace dx$ 🔧
$$\int x^n\thinspace dx\thinspace=\frac{x^{x+1}}{x+1}+C,\medspace n\ne 1$$
$$\thinspace\int x^{-1}\thinspace dx=\ln|x|+C$$
Proof of $\int x^n\thinspace dx$
Proof of $\int (1/x)\thinspace dx$
6.3.2 $\int b^{a\cdot x}\thinspace dx\thinspace$ & $\thinspace\int e^{a\cdot x}\thinspace dx$ 🔧
$$\int b^{a\cdot x}\thinspace dx\thinspace=\frac{b^{a\cdot x}}{a\cdot\ln(b)}+C$$
$$b>0\medspace\land\medspace b\ne 1$$
6.3.3 $\int\log_b x\thinspace dx\thinspace$ & $\thinspace\int\ln(x)\thinspace dx$ 🔧
$$\int\log_b x\thinspace dx=\frac{x\cdot\ln(x)-x}{\ln(b)}+C$$
$$\therefore\int\ln(x)\thinspace dx=x\cdot\ln(x)-x+C$$
Proof
Use the change the base of the logarithm for $e$
$$\int\log_b x\thinspace dx=\int\frac{\ln(x)}{\ln(b)}\thinspace dx$$
Apply the constant multiple rule
$$\int\log_b x\thinspace dx=\frac{1}{\ln(b)}\cdot\int\ln(x)\thinspace dx$$
Define parts for integration
$$u=\ln(x)\qquad u'=\frac{1}{x}\thinspace dx$$
$$v=x\qquad\quad\enspace\thinspace v'=1\thinspace dx\thickspace$$
Rewrite the integral
$$\int\log_b x\thinspace dx=\frac{1}{\ln(b)}\cdot\bigg(\ln(x)\cdot x-\int x\cdot\frac{1}{x}\thinspace dx\bigg)$$
Simplify
$$\int\log_b x\thinspace dx=\frac{1}{\ln(b)}\cdot\bigg(\ln(x)\cdot x-\int dx\bigg)$$
Evaluate with the power rule
