# Integrals of Exponential and Logarithmic Functions

Functions: $${e^x},$$ $${a^x},$$ $$\ln x,$$ $$\sin x,$$ $$\cos x$$
Argument (independent variable): $$x$$
Natural number: $$n$$
Real numbers: $$C$$, $$a$$, $$b$$
1. Integral of the exponential function
$${\large\int\normalsize} {{e^x}dx} = {e^x} + C$$
2. Integral of the exponential function with base a
$${\large\int\normalsize} {{a^x}dx} = {\large\frac{{{a^x}}}{{\ln a}}\normalsize} + C,$$ $$a \gt 0.$$
3. $${\large\int\normalsize} {{e^{ax}}dx} = {\large\frac{{{e^{ax}}}}{{a}}\normalsize} + C,$$ $$a \ne 0.$$
4. $${\large\int\normalsize} {x{e^{ax}}dx} =$$ $${\large\frac{{{e^{ax}}}}{{{a^2}}}\normalsize}\left( {ax – 1} \right) + C,$$ $$a \ne 0.$$
5. Integral of the natural logarithm
$${\large\int\normalsize} {\ln x\,dx} =$$ $$x\ln x – x + C$$
6. $${\large\int\normalsize} {\large\frac{{dx}}{{x\ln x}}\normalsize} =$$ $$\ln \left| {\ln x} \right| + C$$
7. $${\large\int\normalsize} {{x^n}\ln x\,dx} =$$ $${x^{n + 1}}\left[ {{\large\frac{{\ln x}}{{n + 1}}\normalsize} – {\large\frac{1}{{{{\left( {n + 1} \right)}^2}}}}\normalsize} \right]$$ $$+\; C$$
8. $${\large\int\normalsize} {{e^{ax}}\sin {bx}\,dx} =$$ $${\large\frac{{a\sin {bx} – b\cos {bx}}}{{{a^2} + {b^2}}}\normalsize} {e^{ax}} + C$$
9. $${\large\int\normalsize} {{e^{ax}}\cos {bx}\,dx} =$$ $${ \large\frac{{a\cos {bx} + b\sin {bx}}}{{{a^2} + {b^2}}}\normalsize} {e^{ax}} + C$$