## Taylor series sinx

Input interpretation |
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series | sin(x) |

Series expansion at x=0 |

x – x^3/6 + x^5/120 + O(x^7) (Taylor series) |

Approximations about x=0 up to order 3 |

(order n approximation shown with n dots) |

Category: Arc Length

Input interpretation |
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series | sin(x) |

Series expansion at x=0 |

x – x^3/6 + x^5/120 + O(x^7) (Taylor series) |

Approximations about x=0 up to order 3 |

(order n approximation shown with n dots) |

Indefinite integral |
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integral e^(x^2) dx = 1/2 sqrt(pi) erfi(x) + constant |

Alternate form of the integral |

e^(x^2) F(x) + constant |

Series expansion of the integral at x=0 |

x + x^3/3 + x^5/10 + O(x^6) (Taylor series) |

Series expansion of the integral at x=∞ |

1/2 (e^(x^2) (1/x + 1/(2 x^3) + 3/(4 x^5) + O((1/x)^6)) – i sqrt(pi)) |

Indefinite integral |
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integral cos(x) dx = sin(x) + constant |

Alternate form of the integral |

1/2 i e^(-i x) – 1/2 i e^(i x) + constant |

Series expansion of the integral at x=0 |

x – x^3/6 + x^5/120 + O(x^6) (Taylor series) |

Definite integral |

integral_0^(pi/2) cos(x) dx = 1 |

Definite integral mean square |

integral_0^(2 pi) (cos^2(x))/(2 pi) dx = 1/2 = 0.5 |

Indefinite integral |
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integral sin(x) dx = -cos(x) + constant |

Alternate form of the integral |

-1/2 e^(-i x) – e^(i x)/2 + constant |

Series expansion of the integral at x=0 |

-1 + x^2/2 – x^4/24 + O(x^6) (Taylor series) |

Definite integral over a half-period |

integral_0^pi sin(x) dx = 2 |

Definite integral mean square |

integral_0^(2 pi) (sin^2(x))/(2 pi) dx = 1/2 = 0.5 |

Indefinite integral |
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integral 1/x dx = log(x) + constant |

Input interpretation |
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arc length |

Input values |

curve | sin(x) lower limit | 1 upper limit | 2 |

Result |

integral_1^2 sqrt(1 + cos^2(x)) dx~~1.04025… |

Plot |

Input interpretation |
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parametric plane curves | arc length |

Result |

astroid | s = 6 a bifolium | s = 1/6 a (sqrt(4 sqrt(2) – 5) (4 sqrt(2) (K(1/7 (-57 – 40 sqrt(2))) + F(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 5 E(1/7 (-57 – 40 sqrt(2))) + 5 E(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 8/7 (8 + 5 sqrt(2))) cardioid | s = 8 a Cayley sextic | s = 6 pi a cycloid of Ceva | s = a (-3 K(13/16) + 16 E(13/16) + 3 Pi(1/4|13/16)) circle | s = 2 pi a circle parallel curve | s = 2 pi (a + k) circular arc | s = a p cornoid | s = 4 a (-3 K(-2) + E(-2) + 3 Pi(1/4|-2)) deltoid | s = (16 a)/3 |

Example plots |

Input interpretation |
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arc length |

Input values |

curve | sin(x) lower limit | 1 upper limit | 2 |

Result |

integral_1^2 sqrt(1 + cos^2(x)) dx~~1.04025… |

Plot |

Input interpretation |
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parametric plane curves | arc length |

Result |

astroid | s = 6 a bifolium | s = 1/6 a (sqrt(4 sqrt(2) – 5) (4 sqrt(2) (K(1/7 (-57 – 40 sqrt(2))) + F(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 5 E(1/7 (-57 – 40 sqrt(2))) + 5 E(sin^(-1)(1/7 (9 – 4 sqrt(2)))|1/7 (-57 – 40 sqrt(2)))) + 8/7 (8 + 5 sqrt(2))) cardioid | s = 8 a Cayley sextic | s = 6 pi a cycloid of Ceva | s = a (-3 K(13/16) + 16 E(13/16) + 3 Pi(1/4|13/16)) circle | s = 2 pi a circle parallel curve | s = 2 pi (a + k) circular arc | s = a p cornoid | s = 4 a (-3 K(-2) + E(-2) + 3 Pi(1/4|-2)) deltoid | s = (16 a)/3 |

Example plots |