Integration of Exponential and log functions
1. Integral of the exponential function
$$ \int e^x \ dx=e^x + C $$
2. Integral of the exponential function with base a
$$ \int a^x \ dx= \frac {a^x}{\ln a} +C, \ a \gt 0 $$
$$ 3. \ \int e^{ax}dx= \frac {e^{ax}}{a} +C, \ a \neq 0 $$
$$ 4. \ \int xe^{ax}dx= \frac {e^{ax}}{a^2} (ax−1)+C, \ a \neq 0. $$
5. Integral of the natural logarithm
$$ \int \ln x \ dx= x \ln x−x+C $$
$$ 6. \ \int \frac {dx}{x \ln x}= \ln |\ln x| + C $$
$$ 7. \ \int x^n \ln x \ dx= x^{n+1} \left[ \frac {\ln x}{n+1}− \frac {1}{(n+1)^2} \right] + C $$
$$ 8. \ \int e^{ax} \sin bx \ dx= \frac {a\sin bx−b \cos bx}{a^2+b^2} e^{ax} + C $$
$$ 9. \ \int e^{ax} \cos bx \ dx= \frac {a\cos bx+b \sin bx}{a^2+b^2} e^{ax} + C $$
$$ 10. \ \int \log x \ dx = \frac {(x \cdot \ln x – x)}{\ln (10)} + C $$
$$ 11. \ \int \log_ax dx = x(\log ax – \log ae) + C $$
Example:
$$ \text{Integrate} \int e^{3x}dx $$
Solution:
$$ \text{Let} \ u=3x, \text{Then} \ du=3dx $$
$$ \int e^{3x}dx= \frac 13 \int e^udu $$
$$ = \frac 13 e^u + C $$
$$ = \frac {e^{3x}}{3} +C $$