# Trapezoidal Rule Calculator

Calculator will approximate integral using trapezoidal rule.

- In general, you can skip multiplication sign, so `5x` is equivalent to `5*x`
- In general, you can skip parentheses, but be very careful: e^3x is `e^3x` and e^(3x) is `e^(3x)`
- Also be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)` and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`
- If you skip parentheses or multiplication sign, type at least whitespace, i.e. write sin x (or even better sin(x)) instead of sinx
- Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x)
- Similarly tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x)
- From table below you can notice, that sech is not supported, but you can still enter it using identity `sech(x)=1/cosh(x)`
- If you get an error, double check your expression, add parentheses and multiplication signs, where needed, and consult the table below
- All suggestions and improvements are welcomed. Leave them in comments

Type | Get |

Constants | |

e | e |

pi | `pi` |

i | i (imaginary unit) |

Operations | |

a+b | a+b |

a-b | a-b |

a*b | `a*b` |

a^b, a**b | `a^b` |

sqrt(x), x^(1/2) | `sqrt(x)` |

cbrt(x), x^(1/3) | `root(3)(x)` |

root(x,n), x^(1/n) | `root(n)(x)` |

x^(a/b) | `x^(a/b)` |

abs(x) | `|x|` |

Functions | |

e^x | `e^x` |

ln(x), log(x) | ln(x) |

ln(x)/ln(a) | `log_a(x)` |

Trigonometric Functions | |

sin(x) | sin(x) |

cos(x) | cos(x) |

tan(x) | tan(x), tg(x) |

cot(x) | cot(x), ctg(x) |

sec(x) | sec(x) |

csc(x) | csc(x), cosec(x) |

Inverse Trigonometric Functions | |

asin(x), arcsin(x), sin^-1(x) | asin(x) |

acos(x), arccos(x), cos^-1(x) | acos(x) |

atan(x), arctan(x), tan^-1(x) | atan(x) |

acot(x), arccot(x), cot^-1(x) | acot(x) |

asec(x), arcsec(x), sec^-1(x) | asec(x) |

acsc(x), arccsc(x), csc^-1(x) | acsc(x) |

Hyperbolic Functions | |

sinh(x) | sinh(x) |

cosh(x) | cosh(x) |

tanh(x) | tanh(x) |

coth(x) | coth(x) |

1/cosh(x) | sech(x) |

1/sinh(x) | csch(x) |

Inverse Hyperbolic Functions | |

asinh(x), arcsinh(x), sinh^-1(x) | asinh(x) |

acosh(x), arccosh(x), cosh^-1(x) | acosh(x) |

atanh(x), arctanh(x), tanh^-1(x) | atanh(x) |

acoth(x), arccoth(x), cot^-1(x) | acoth(x) |

acosh(1/x) | asech(x) |

asinh(1/x) | acsch(x) |

## Solution

**Your input: approximate integral $$$\int_{0}^{1}\sqrt{\sin^{3}{\left (x \right )} + 1}\ dx$$$ using $$$n=5$$$ rectangles**.

Trapezoidal rule states that $$$\int_{a}^{b}f(x)dx\approx\frac{\Delta{x}}{2}\left(f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)\right)$$$, where $$$\Delta{x}=\frac{b-a}{n}$$$.

We have that $$$a=0$$$, $$$b=1$$$, $$$n=5$$$.

Therefore, $$$\Delta{x}=\frac{1-0}{5}=\frac{1}{5}$$$.

Divide interval $$$\left[0,1\right]$$$ into $$$n=5$$$ subintervals of length $$$\Delta{x}=\frac{1}{5}$$$ with the following endpoints: $$$a=0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1=b$$$.

Now, we just evaluate function at those endpoints:

$$$f\left(x_{0}\right)=f(a)=f\left(0\right)=1=1$$$

$$$2f\left(x_{1}\right)=2f\left(\frac{1}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{1}{5} \right )} + 1}=2.00782606791279$$$

$$$2f\left(x_{2}\right)=2f\left(\frac{2}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{2}{5} \right )} + 1}=2.05820697233265$$$

$$$2f\left(x_{3}\right)=2f\left(\frac{3}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{3}{5} \right )} + 1}=2.17257446116512$$$

$$$2f\left(x_{4}\right)=2f\left(\frac{4}{5}\right)=2 \sqrt{\sin^{3}{\left (\frac{4}{5} \right )} + 1}=2.34021475342487$$$

$$$f\left(x_{5}\right)=f(b)=f\left(1\right)=\sqrt{\sin^{3}{\left (1 \right )} + 1}=1.26325897447473$$$

Finally, just sum up above values and multiply by $$$\frac{\Delta{x}}{2}=\frac{1}{10}$$$: $$$\frac{1}{10}(1+2.00782606791279+2.05820697233265+2.17257446116512+2.34021475342487+1.26325897447473)=1.08420812293102$$$

**Answer: 1.08420812293102**.