# Integrals of Trigonometric Functions

Trigonometric functions: $$\sin x,$$ $$\cos x,$$ $$\tan x,$$ $$\cot x,$$ $$\arcsin x,$$ $$\arccos x,$$ $$\arctan x,$$ $$\text {arccot }x$$
Argument (independent variable): $$x$$
Natural numbers: $$m$$, $$n$$
Real number: $$C$$
1. Integral of sine
$${\large\int\normalsize} {\sin x\,dx} = – \cos x + C$$
2. Integral of cosine
$${\large\int\normalsize} {\cos x\,dx} = \sin x + C$$
3. Integral of sine squared
$${\large\int\normalsize} {{{\sin }^2}x\,dx} =$$ $${\large\frac{x}{2}\normalsize} – {\large\frac{1}{4}\normalsize} \sin{2x} + C$$
4. Integral of cosine squared
$${\large\int\normalsize} {{{\cos }^2}x\,dx} =$$ $${\large\frac{x}{2}\normalsize} + {\large\frac{1}{4}\normalsize} \sin{2x} + C$$
5. Integral of sine cubed
$${\large\int\normalsize} {{{\sin }^3}x\,dx} =$$ $${\large\frac{1}{3}\normalsize}{\cos ^3}x – \cos x + C =$$ $${\large\frac{1}{{12}}\normalsize}\cos {3x} – {\large\frac{3}{4}\normalsize} \cos x + C$$
6. Integral of cosine cubed
$${\large\int\normalsize} {{{\cos }^3}x\,dx} =$$ $$\sin x – {\large\frac{1}{3}\normalsize}{\sin ^3}x + C =$$ $${\large\frac{1}{{12}}\normalsize}\sin {3x} + {\large\frac{3}{4}\normalsize} \sin x + C$$
7. Integral of secant
$${\large\int {\frac{{dx}}{{\cos x}}}\normalsize} =$$ $${\large\int\normalsize} {\sec x\,dx} =$$ $$\ln \left| {\tan \left( {{\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}\normalsize}} \right)} \right| + C =$$ $${\ln \left| {\sec x + \tan x} \right| + C}$$
8. Integral of cosecant
$${\large\int {\frac{{dx}}{{\sin x}}}\normalsize} =$$ $${\large\int\normalsize}{\csc x\,dx} =$$ $$\ln \left| {\tan {\large\frac{x}{2}\normalsize}} \right| + C$$ = $${-\ln \left| {\csc x + \cot x} \right| + C}$$
9. Integral of secant squared
$${\large\int {\frac{{dx}}{{{{\cos }^2}x}}}\normalsize} =$$ $${\large\int\normalsize} {{{\sec }^2}x\,dx} =$$ $$\tan x + C$$
10. Integral of cosecant squared
$${\large\int {\frac{{dx}}{{{{\sin }^2}x}}}\normalsize} =$$ $${\large\int\normalsize} {{{\csc }^2}x\,dx} =$$ $$-\cot x + C$$
11. Integral of secant cubed
$${\large\int {\frac{{dx}}{{{{\cos }^3}x}}}\normalsize} =$$ $${\large\int\normalsize} {{{\sec }^3}xdx} =$$ $${\large\frac{{\sin x}}{{2{{\cos }^2}x}}\normalsize}$$ $$+\;{\large\frac{1}{2}\normalsize}\ln \left| {\tan \left( {{\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}}\normalsize} \right)} \right|$$ $$+\;C$$
12. Integral of cosecant cubed
$${\large\int {\frac{{dx}}{{{{\sin }^3}x}}}\normalsize} =$$ $${\large\int\normalsize} {{{\csc }^3}xdx} =$$ $$-{\large\frac{{\cos x}}{{2{{\sin }^2}x}}\normalsize}$$ $$+\;{\large\frac{1}{2}\normalsize}\ln \left| {\tan {\large\frac{x}{2}\normalsize}} \right|$$ $$+\; C$$
13. Integral of the product of sine and cosine
$${\large\int\normalsize} {\sin x\cos x\,dx} =$$ $$– {\large\frac{1}{4}\normalsize}\cos{2x} + C$$
14. Integral of the product of sine squared and cosine
$${\large\int\normalsize} {{{\sin }^2}x\cos x \,dx} =$$ $${\large\frac{1}{3}\normalsize}{\sin^3}x + C$$
15. Integral of the product of cosine squared and sine
$${\large\int\normalsize} {{{\cos }^2}x\sin x \,dx} =$$ $$-{\large\frac{1}{3}\normalsize} {\cos^3}x + C$$
16. Integral of the product of sine squared and cosine squared
$${\large\int\normalsize} {{{\sin }^2}x\,{{\cos }^2}x\,dx} =$$ $${\large\frac{x}{8}\normalsize} – {\large\frac{1}{{32}}\normalsize} \sin{4x} + C$$
17. Integral of tangent
$${\large\int\normalsize} {\tan x\,dx} =$$ $$- \ln \left| {\cos x} \right| + C$$
18. $${\large\int\normalsize} {{\large\frac{{\sin x}}{{{{\cos }^2}x}}\normalsize} dx} =$$ $${\large\frac{1}{{\cos x}}\normalsize} + C =$$ $$\sec x + C$$
19. $${\large\int\normalsize} {{\large\frac{{{{\sin }^2}x}}{{\cos x}}\normalsize} dx} =$$ $$\ln \left| {\tan \left( {{\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}\normalsize}} \right)} \right|$$ $$-\;\sin x + C$$
20. Integral of tangent squared
$${\large\int\normalsize} {{{\tan }^2}x\,dx} =$$ $$\tan x – x + C$$
21. Integral of cotangent
$${\large\int\normalsize} {\cot x\,dx} =$$ $$\ln \left| {\sin x} \right| + C$$
22. $${\large\int\normalsize} {{\large\frac{{\cos x}}{{{{\sin }^2}x}}\normalsize} dx} =$$ $$– {\large\frac{1}{{\sin x}}\normalsize} + C =$$ $$– \csc x + C$$
23. $${\large\int\normalsize} {{\large\frac{{{\cos^2}x}}{{\sin x}}\normalsize} dx} =$$ $$\ln \left| {\tan {\large\frac{x}{2}}\normalsize} \right|$$ $$+\;\cos x + C$$
24. Integral of cotangent squared
$${\large\int\normalsize} {{{\cot }^2}x\,dx} =$$ $$– \cot x – x + C$$
25. $${\large\int\normalsize} {\large\frac{{dx}}{{\cos x\sin x}}\normalsize} =$$ $$\ln \left| {\tan x} \right| + C$$
26. $${\large\int\normalsize} {\large\frac{{dx}}{{{\sin^2}x\cos x}}\normalsize} =$$ $$– {\large\frac{1}{{\sin x}}\normalsize} + \ln \left| {\tan \left( {{\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}}\normalsize} \right)} \right|$$ $$+\;C$$
27. $${\large\int\normalsize} {\large\frac{{dx}}{{\sin x\,{{\cos }^2}x}}\normalsize} =$$ $${\large\frac{1}{{\cos x}}\normalsize}$$ $$+\; \ln \left| {\tan {\large\frac{x}{2}}\normalsize} \right| + C$$
28. $${\large\int {\frac{{dx}}{{{\sin^2}x\,{{\cos }^2}x}}}\normalsize} =$$ $$\tan x – \cot x + C$$
29. $${\large\int\normalsize} {\sin {mx}\sin {nx}\,dx} =$$ $$– {\large\frac{{\sin \left( {m + n} \right)x}}{{2\left( {m + n} \right)}}\normalsize}$$ $$+\; {\large\frac{{\sin \left( {m – n} \right)x}}{{2\left( {m – n} \right)}}\normalsize} + C,$$ $${m^2} \ne {n^2}.$$
30. $${\large\int\normalsize} {\sin {mx}\cos {nx}\,dx} =$$ $$– {\large\frac{{\cos \left( {m + n} \right)x}}{{2\left( {m + n} \right)}}\normalsize}$$ $$-\;{\large\frac{{\cos \left( {m – n} \right)x}}{{2\left( {m – n} \right)}}\normalsize} + C,$$ $${m^2} \ne {n^2}.$$
31. $${\large\int\normalsize} {\cos {mx}\cos {nx}\,dx} =$$ $${\large\frac{{\sin \left( {m + n} \right)x}}{{2\left( {m + n} \right)}}\normalsize}$$ $$+\;{\large\frac{{\sin \left( {m – n} \right)x}}{{2\left( {m – n} \right)}}\normalsize} + C,$$ $${m^2} \ne {n^2}.$$
32. $${\large\int\normalsize} {\sec x\tan x\,dx} =$$ $$\sec x + C$$
33. $${\large\int\normalsize} {\csc x\cot x\,dx} =$$ $$-\csc x + C$$
34. $${\large\int\normalsize} {\sin x\,{{\cos }^n}x\,dx} =$$ $$– {\large\frac{{{{\cos }^{n + 1}}x}}{{n + 1}}\normalsize} + C$$
35. $${\large\int\normalsize} {{{\sin }^n}x\cos x\,dx} =$$ $${\large\frac{{{\sin^{n + 1}}x}}{{n + 1}}\normalsize} + C$$
36. Integral of inverse sine
$${\large\int\normalsize} {\arcsin x\,dx} =$$ $$x\arcsin x$$ $$+\; \sqrt {1 – {x^2}} + C$$
37. Integral of inverse cosine
$${\large\int\normalsize} {\arccos x\,dx} =$$ $$x\arccos x$$ $$-\; \sqrt {1 – {x^2}} + C$$
38. Integral of inverse tangent
$${\large\int\normalsize} {\arctan x\,dx} =$$ $$x\arctan x$$ $$-\; {\large\frac{1}{2}\normalsize}\ln \left( {{x^2} + 1} \right) + C$$
39. Integral of inverse cotangent
$${\large\int\normalsize} {\text {arccot }x\,dx} =$$ $$x\,\text{arccot }x$$ $$+\; {\large\frac{1}{2}\normalsize}\ln \left( {{x^2} + 1} \right) + C$$