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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
      \({\large\int\normalsize} {\text {arccot }x\,dx} =\) \( x\,\text{arccot }x \) \(+\; {\large\frac{1}{2}\normalsize}\ln \left( {{x^2} + 1} \right) + C\)