Integration and Differentiation Formulas PDF
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This document contains essential integration and differentiation formulas and rules. It lists basic properties, formulas, common derivatives for polynomials, trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions. It's a helpful resource for students studying calculus.
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# Revision Rules 1. ∫ f(x) + g(x)dx = ∫ f(x)dx + ∫ g(x)dx 2. ∫ cf(x)dx = c ∫ f(x)dx 3. ∫ Kdx = Kx + C (CEIR) 4. ∫ x^n dx = x^(n+1) / (n+1) + C 5. ∫ 1/x dx = ln|x| + C 6. ∫ e^kx dx = e^kx / k + C 7. ∫ b^x dx = b^x / ln(b) + C 8. ∫ sin(kx)dx = -cos(kx) / k + C 9. ∫ cos(kx)dx = sin(kx)...
# Revision Rules 1. ∫ f(x) + g(x)dx = ∫ f(x)dx + ∫ g(x)dx 2. ∫ cf(x)dx = c ∫ f(x)dx 3. ∫ Kdx = Kx + C (CEIR) 4. ∫ x^n dx = x^(n+1) / (n+1) + C 5. ∫ 1/x dx = ln|x| + C 6. ∫ e^kx dx = e^kx / k + C 7. ∫ b^x dx = b^x / ln(b) + C 8. ∫ sin(kx)dx = -cos(kx) / k + C 9. ∫ cos(kx)dx = sin(kx) / k + C 10. ∫ cosec^2(kx)dx = -cot(kx) / k + C 11. ∫ sec^2(kx)dx = tan(kx) / k + C 12. ∫ cosec(kx)cot(kx)dx = -cosec(kx) / k + C 13. ∫ sec(kx)tan(kx)dx = sec(kx) / k + C 14. ∫ cosec^2(kx)dx = -cot(kx) / k + C 15. ∫ 1 / (1 + x^2) dx = tan^-1(x) + C 16. ∫ 1 / (√(1 - x^2)) dx = sin^-1(x) + C 17. ∫ tan(x) dx = ln|sec(x)| + C or -ln|cos(x)| + C 18. ∫ cot(x) dx = ln|sin(x)| + C 19. ∫ sec(x) dx = ln|sec(x) + tan(x)| + C 20. ∫ cosec(x) dx = ln|cosec(x) - cot(x)| + C 21. ∫ 1 / (a^2 + x^2) dx = 1 / a tan^-1(x / a) + C # Derivatives ## Basic Properties/Formulas/Rules * d/dx(cf(x)) = cf'(x), c is any constant. * d/dx(x^n) = nx^(n-1), n is any number. ## Common Derivatives | Derivative | Result | |---|---| | d/dx(e^(g(x)) | g'(x)e^(g(x))| | d/dx(ln(g(x)) | g'(x) / g(x) | ### Polynomials | Derivative | Result | |---|---| | d/dx(c) | 0 | | d/dx(x) | 1 | | d/dx(cx) | c | | d/dx(x^n) | nx^(n-1) | | d/dx(cx^n) | nc x^(n-1) | ### Trig Functions | Derivative | Result | |---|---| | d/dx(sin(x)) | cos(x) | | d/dx(cos(x)) | -sin(x) | | d/dx(tan(x)) | sec^2(x) | | d/dx(csc(x)) | -csc(x)cot(x) | | d/dx(sec(x)) | sec(x)tan(x) | | d/dx(cot(x)) | -csc^2(x) | ### Inverse Trig Functions | Derivative | Result | |---|---| | d/dx(sin^-1(x)) | 1 / √(1 - x^2) | | d/dx(cos^-1(x)) | -1 / √(1 - x^2) | | d/dx(tan^-1(x)) | 1 / (1 + x^2) | | d/dx(csc^-1(x)) | -1 / (|x|√(x^2 - 1)) | | d/dx(sec^-1(x)) | 1 / (|x|√(x^2 - 1)) | | d/dx(cot^-1(x)) | -1 / (1 + x^2) | ### Exponential & Logarithm Functions | Derivative | Result | |---|---| | d/dx(a^x) | a^x ln(a) | | d/dx(e^x) | e^x | | d/dx(ln(x)) | 1 / x, x > 0 | | d/dx(1/x) | -1 / x^2, x ≠ 0 | | d/dx(log_a(x)) | 1 / (xln(a)), x > 0 |
