\[ \int e^x dx = e^x \]

\[ \int a^x dx = {a^x \over \ln a}, \quad a \gt 0, a \neq 1 \]

\[ \int e^{ax} dx = \frac{1}{a} e^{ax}, \quad a \neq 0 \]

\[ \int x\,e^{ax} dx = \frac{ax-1}{a^2} e^{ax}, \quad a \neq 0 \]

\[ \int x^2 e^{ax} dx = \frac{a^2 x^2-2ax+2}{a^3} e^{ax}, \quad a \neq 0 \]

\[ \int x^3 e^{ax} dx = \frac{a^3 x^3-3a^2 x^2+6ax-6}{a^4} e^{ax}, \quad a \neq 0 \]

\[ \int x\,e^{x^2} dx = \frac12 e^{x^2} \]

\[ \int x^3 e^{x^2} dx = \frac12 (x^2-1) e^{x^2} \]

\[ \int \sqrt{x}\,e^{\sqrt{x}} dx = 2(x-2\sqrt{x}+2) e^{\sqrt{x}} \]

\[ \int {e^{\sqrt{x}} \over \sqrt{x}} dx = 2 e^{\sqrt{x}} \]

\[ \int {dx \over e^x + 1} = \ln{e^x \over e^x + 1} \]

\[ \int {dx \over a+be^{cx}} = \frac{x}{a} - \frac{1}{ac} \ln|a+be^{cx}|, \quad a \neq 0, c \neq 0 \]

\[ \int {dx \over (a+be^{cx})^2} = \frac{1}{a^2c} \ln\left|{e^{cx} \over a+be^{cx}}\right| + {1 \over ac(a+be^{cx})}, \quad a \neq 0, c \neq 0 \]

\[ \int {e^{cx} dx \over a+be^{cx}} = \frac{1}{ac} \ln|a+be^{cx}|, \quad a \neq 0, c \neq 0 \]

\[ \int {dx \over \sqrt{e^x - 1}} = 2\, \mathtt{arctg}\sqrt{e^x - 1} \]

\[ \int e^{ax} \sin bx \, dx = {a \sin bx - b \cos bx \over a^2+b^2} e^{ax}, \quad a^2+b^2 \neq 0 \]

\[ \int e^{ax} \cos bx \, dx = {a \cos bx + b \sin bx \over a^2+b^2} e^{ax}, \quad a^2+b^2 \neq 0 \]

\[ \int e^{ax} \sin^2 x \, dx = {e^{ax} \over a^2+4} \left(a \sin^2 x - \sin 2x + \frac{2}{a}\right), \quad a \neq 0 \]

\[ \int e^{ax} \cos^2 x \, dx = {e^{ax} \over a^2+4} \left(a \cos^2 x + \sin 2x + \frac{2}{a}\right), \quad a \neq 0 \]