Powers
\[ 1. \int \frac{1}{x} dx = ln(x) + C \]
\[ 2. \int x^a dx = \frac{x^{a+1}}{a+1} + C \]
\[ 3. \int a^x dx = \frac{a^x}{ln(a)} + C \]
\[ 4. \int \frac{1}{x^a} dx = -\frac{x^{(1-a)}}{1-a} + C \]
Trigonometrics
\[ 1. \int sin(x) dx = -cos(x) + C \]
\[ 2. \int \frac{1}{sin(x)} dx = ln\Bigg|\frac{1}{sin(x)} - \frac{cos(x)}{sin(x)}\Bigg| + C \]
\[ 3. \int sin(ax)sin(bx) dx = \frac{sin(a-b)x}{2(a-b)} - \frac{sin(a+b)x}{2(a+b)} + C \]
\[ 4. \int sin^2(x) dx = \frac{1}{2}x - \frac{1}{4}sin(2x) + C \]
\[ 5. \int sin^3(x) dx = \frac{1}{12}\Big(cos(3x)-9cos(x)\Big) + C \]
\[ 6. \int cos(x) dx = sin(x) + C \]
\[ 7. \int \frac{1}{cos(x)} dx = ln|sec(x)+tan(x)| + C \]
\[ 8. \int cos(ax)cos(bx) dx = \frac{sin(a-b)x}{2(a-b)} + \frac{sin(a+b)x}{2(a+b)} + C \]
\[ 9. \int cos^2(x) dx = \frac{1}{2}x + \frac{1}{4}sin(2x) + C \]
\[ 10. \int cos^3(x) dx = \frac{1}{12}\Big(9sin(x)+sin(3x)\Big) + C \]
\[ 11. \int sin(x)cos(x) dx = -\frac{cos^2(x)}{2} + C \]
\[ 12. \int sin^2(x)cos^2(x) dx = \frac{1}{32}\Big(4x-sin(4x)\Big) + C \]
\[ 13. \int tan(x) dx = -ln|cos(x)| + C \]
\[ 14. \int tan(x) dx = ln|sec(x)| + C \]
\[ 15. \int tan^2(x) dx = tan(x)-x + C \]
\[ 16. \int tan^3(x) dx = \frac{tan^2(x)}{2} + ln|cos(x)| + C \]
\[ 17. \int \frac{1}{tan(x)} dx = ln|sin(x)| + C \]
\[ 18. \int \frac{1}{tan^2(x)} dx = -\frac{1}{tan(x)}-x + C \]
\[ 19. \int \frac{1}{tan^3(x)} dx = \frac{tan^2(x)}{2} - ln|sin(x)| + C \]
Reciprocals
\[ 1. \int \frac{1}{a^2+x^2} dx = \frac{1}{a} arctan\Big(\frac{x}{a}\Big) + C \]
\[ 2. \int \frac{1}{x^2-a^2} dx = \frac{1}{a} ln(a^2+x^2) + C \]
\[ 3. \int \frac{1}{1-x^2} dx = \frac{ln|u+1|-ln|u-1|}{2} + C \]
\[ 4. \int \frac{1}{a^2-x^2} dx = \frac{1}{2a} ln\Bigg|\frac{x+a}{x-a}\Bigg| + C \]
\[ 5. \int \frac{1}{x^2-a^2} dx = \frac{1}{2a} ln\Bigg|\frac{x-a}{x+a}\Bigg| + C \]
\[ 6. \int \frac{1}{\sqrt{a^2+u^2}} dx = ln(u+\sqrt{a^2+x^2}) + C \]